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EXERCISES IN ALGEBHA
BY
EDWARD R. ROBBINS
M
AND
FREDERICK H. SOMERVILLE
WILLIAM PENN CHARTER SCHOOL, PHILADELPHIA
NEW YORK •:• CINCINNATI •:• CHICAGO
AMERICAN BOOK COMPANY
Copyright, 1904, by
EDWARD R. BOBBINS and FREDERICK H. SOMERVILLE.
Enteeed at Stationers' Hall, London.
K. A 8. EXERCISES IN ALG.
W. P. I
Tit.
PEEFACE
The present-day teacher of Algebra has little time for the
selection, from proper sources, much less for the making, of
suitable examples often necessary to supplement those con-
tained in the standard text-books. This little book is designed
to meet the requirements of those teachers who feel such extra
assignments in Algebra essential to thorough familiarity with
its processes. The aim has been to provide, as compactly as
possible, a series of exercises that conform in arrangement with
the order of the leading text-books, and that both in degree
of difficulty and in scope shall include the work prescribed by
high schools and academies, as well as university and college
entrance requirements.
The plan has been to avoid all examples of more than aver-
age difficulty, and to lay particular stress upon those subjects
that stand so clearly as the foundations of later work. With
this in view, much emphasis has been given to the subjects of
Factoring, Fractions, Exponents, Equations, and Logarithms.
There has been constant effort to present abundant drill in
topics in the mastering of which students seem to have the
greatest difficulties.
Not only must the pupil who is to master the science solve
a multitude of exercises differing in degree rather than in
S
4 PREFACE
kind, but he must also be taught to select appropriate methods
for each of a miscellaneous collection, differing in kind rather
than in degree. To this end the book is generously provided
with reviews, and the pupil is obliged to discriminate among
different kinds of examples — a phase of mathematical train-
ing which will be invaluable to him in subsequent study.
Furthermore, the reviews are so arranged that they could, if
desired, be used independently of the other exercises in final
review, or in final preparation for college examinations.
The authors will welcome and will appreciate any sugges-
tions or corrections from other teachers of Algebra.
EDWARD R. ROBBINS.
FREDERICK H. SOMERVILLE.
Philadelphia, Pa.
CONTENTS
PAOK
Substitution 7
Addition 9
Subti'action 10
Use of the Parenthesis 13
Review 16
Multiplication 17
Division 18
Multiplication by Inspection 20
Division by Inspection 23
Use of the Parenthesis with Multiplication 25
Simple Equations 26
Problems in Simple Equations 27
Review 29
Factoring 31
Review . 38
Highest Common Factor and Lowest Common Multiple . . 40
Fractions :
Transformations 42
Addition and Subtraction .45
Multiplication and Division 48
Complex Fractions 50
Fractional Equations :
Numerical Equations 55
Literal Equations 59
Simultaneous Equations :
Numerical Equations 61
Literal Equations 64
Three or More Unknown Quantities 66
Problems in Simultaneous Equations 68
6
6 CONTENTS
Involution and Evolution :
Monomials 71
Involution — Binomials 72
Evolution — Square Root .72
Evolution — Cube Root 74
Evolution — Numerical . . . . . . . .74
Review 7G
Exponents :
Transformations 81
Miscellaneous Applications . 88
Radicals :
Transformations . . .93
Miscellaneous Applications 99-
Imaginaries 103
General Review . . 107
Quadratic Equations :
Numerical Quadratics 116
Literal Quadratics 119
Equations in Quadratic Form 121
Simultaneous Quadratics 123
Properties of Quadratics 126
Ratio and Proportion . . . 129
Variation 132
Arithmetical Progression 134
Geometrical Progression 138
Permutations and Combinations 142
Binomial Theorem 145
Logarithms 148
General Review 156
SUBSTITUTION
Exercise 1
Find the numerical value of the following :
When a = 1, 6 = 2, c = 3, d = 4.
1. a + b. 10. 7b-(c + d),
2. a + & + c. 11. a + ab.
3. a-f26 + 3c. 12. 3a6-c.
4. a + 36 — d 13. 12a — 3c + cd
5. 2 a 4- 4 6 — 2d 14. 3a + &(a + c).
6. 6a — 6 — d 15. ab-}-a{2b — a).
7. 10c-56 + 2d 16. 4a6c-3(c + d).
8. 3a + c + d 17. a(a + b + c).
9. 3a4-(c4-d)- 18. a6(a + 6 + c).
19. a6(d-a)4-&c(c?-6).
20. a + ab(b + c)-c(3d-3c),
21. c2-|-a6. 25. ab^d - a(b^ -^ c).
22. 62 4.c2 + cZ2. 26. 25d-a62(a + 6 + c).
23. 2a'b^ + Sb^-G' + d. 27. a + (a + 6)1
24. &V + 6c2 + d 28. 4 6cd + (2 6 + c)2.
29. 3(a + 2)2-2(62_l) + 3a26c2.
30. 2(a-{-by-S(d-by-bc{c-hd).
When a = 5, b =4:,
31. V6 + Ve.
32. Vie— V2d.
SUBSTITUTION
c = S, d = 2, e = 9.
33.
34.
V5 ab + V4 6.
V3 ce — V3 6c.
35.
•y/abc 4- 6.
37.
Va^ 4- 6' + 8.
38.
V2c2 + 3d2-a.
39.
a + b -Vbcd -f 1.
40.
(a + 6)V6cd + l.
36. -Vabc — Sc — d.
41. (ad — e) V 3 ac — e.
42. (a + Ve)-(e- V6).
When a = 4, 6 =
= 5,
m
= 6,
n = 10.
43. ^«
m
19 3a + 26
n — 1
44 ^ + ^
2a
3 a + (w — -wi)
* 2(n-a)
45. ^(^^-^).
a
51. ^4-^ + ^.
m n
46. ^(^ + ^) .
a6
b n 2a
47.
m-\-n — g
m
/ V2a6 + am\ ,
oo. I ; ]-T ^n — m.
\ m + n J
48. ^(^-^) .
Whena; = i 2/ = |,
55. iC+2/ + ^-
56. 2x — y — z.
57. a;(x + ?/).
58. 2/ (2 a; — 2;).
54.
^m
^n—m Via
+ V3
2; =
59. {x-[-y){y-z).
60. aJ2/(a;+?/-2;).
61. {x + lf-{y^-iyH^ + iy.
62. aJ + 2/
("J-
(0^ + ^)^.
ADDITION y
When a = 4, 6=5, m = 2, w = 3.
63. or. 68. (a + 6)" — (a + 6)"*.
64. a'^ + 6". 69. (26-a)~-(36-a)'".
65. a^' — b'^. 70. (a + 6 — 7)"'.
66. 2a'' + 3a"'b\ 71. (a'^ + ft"')".
67. (a + 6)"'. 72. a~ + a'"(6 - a)".
ADDITION
Ii2:ercise 2
1. Find the sum ofa + 36 + c, 2a4-76 + 2c, and 3 a + 2 6
+ c.
2. Find the sum of 4a + 36 — c, 2a4-264-4c, and a —
3 6 -2 c.
3. Findthesumof 10a-36-2c-d, -2a + 4:b-\-c + Sd,
and c — 3 d + a.
4. Add 3a + 26-3c, 12a-46-7c, and 4a-86 + 9c.
5. Add 3a^ + 2a^-2« + ll, 4.a^-2x^-\-3x-S, 4.a^-2x'
-\-x-2, -12ar^ + a;2-x-l, and 2 o^ -\- oF - x -\- IS.
6. Collect 5a-3c + 4/-m + 26-d + 4c-2a-3/+c
+ 2 m + d.
7. Collect 2 a6 4- 3 6c + 4 cd — 2 6c + 3 a6 — 3 cd — 4 a6 -
2 6c-2cd
8. Collect 2a6cd + 3 6fljH-m — 3a6cc2 4-2 6a; — 3m + 6a6cd
— 3 6» + 3 m.
9. Collect a^-hSa^ + Sa, a^ + a + l, and 2a + 2.
10. Collect a^ + a'b-\-ab% 3a^ + a'b^-2b% and 4a62-3 6^
10 SUBTRACTION
11. Arrange in descending powers of x and collect a:^ — 2 +
^x'-x, -a;4-3a^-2ar*, _4ar' + 5a^-a;4- 10, -a;-f4a^-
2 + ic8, and a^ + a;^ _,_ ^ _,_ ^^
12. Arrange in ascending powers of a and add a^ — 4 a^4-
2a-8, a^ + a-3a2 + 16, a34-a'-2 + 3a, and -^.a' + Qa?
+ 10 a - 3.
13. Find the sum ofa^-l+3a;-a^ + 2a;-3a:3 + 44-«^-
8a; + 2a.'^ — 4a^ + 10 — ic^, and arrange the answer in descend-
ing powers of x.
14. Add 3(a4-6)4-4(c + d)-5(w4-n), 2(a + &)-2(c + c?)
-\-Q{m-\- n), and — 4 (a + 6) — (c + d) — (m + n).
15.
16.
17.
18.
O/C
ac
mx
my
be
c
5x
-Sy
{a-\-b)c (a + l)c ( )» ( )y
19. Add ab + cc? and mb + ncZ.
20. Add ax + by and ca; + dy.
21. Add 2 a6 + 3 c 4- c? and 3 6 + oc + nd
22. Add 3a6+2ac+w, 36— 3ac+mw, and 2 nib +4: be— pn.
23. Add (a + &)c + (6 4- (^)c.
24. Add (m + 7i) a; + (m — n) a; — mx.
25. Add (2 + a)a^ + (3a-4)a^+(a-l)ar^-4aar^.
Subtract :
1. 2. 3.
10 a 10 a - 4 a
3 a —4 a 3 a
SUBTRACTION
Exercise 3
4. 5.
-3 a 15a'
-a -2a'
6.
4a6
3a6
7.
3a6
7a6
8.
2 am
—6 am
SUBTRACTIOn
11
9.
10. 11.
12.
13.
3aH-26
14a + 3c 16a-2
c 12 aft + 16
— 5a — 4 c
a+ h
7a— c 3a + 5
c 3 a6 - 10
-2a-5c
14.
15.
16.
17.
a^ - 10 ah
2 a^ + 12 a?7i
5a6
16 am 4- mn
a^H- 3a6
3a2+ 3 am
4a6 + cd
Sam
18.
19.
20.
4 a^c 4- 3 a6 + 10 15 a?
10 a^ + 3
! a^ 4- 6 a + 7
-2 ah
10a2 + 4a
+ 4
a^ -4
21. Take a^h + c from 4a + 364-2c.
22. Take 5 a — 6 4- c from 6 a + 3 6 — 4 c.
23. Take 2a4-36— 4c from 4a4-36H-8c. ,
24. Take 3 a + 10 6 - 14 c from 4 a 4- 10 6 - 14 c.
25. Take a- + 3 a6 + 12 from 5 a^ - 8 a6 + 16.
26. Take 4a2 4- 3a6 4-2a&2_ 10 from 8a2-16a6-3a62_l.
27. Take 3a2 4-3a + l from 4a3 + 5a2-6a-3.
28. From 3 a^- 2 a^ + e a2 4-2 a-3 take 2a^-a3-4a2-
a + 1.
29. From a* - 4 a^ft + 6 a^h' - 4 aW+h"^ take a36+3 a%^+ah\
30. From -3a2 4-8a + 36 take a2-2a-18.
31. From -16a3-8a2 4-4a-5 take -2 a^ + a'^ -a-\-l.
32. Given a minuend a^4-4a- — 3a4-2 and a subtrahend
a^ 4- 3 a^ — 4 a 4- 1, find the difference.
33. Given a subtrahend — 3 ac, a minuend 8 ac, find differ-
ence.
12 BUBTRACTION
34. Given a minuend 3 ahc, a difference 4 ahc, find subtra-
hend.
35. Given a minuend axy, a difference —axy, find subtra-
hend.
36. What must be added to 16 ac to make —Sac?
37. What must be added to 75 m to make 31 m ?
38. What must be added to — 3 a^ to make ?
39. 36 a^ is added to what expression to make 82 a^?
40. What expression added to 4 a -f- 2 c will give 5 a + 8 c^ ?
41 . Given a subtrahend 4a'*4-3a^ — 2a^ + a — 7 and a differ-
ence — 3 a^ + 2 a^ — a, find the minuend.
42. Given a minuend 8m^ — 3 mn + 2 n^-f 18 and a differ-
ence 2 m^ + 2 mn — n^-\- 16, find the subtrahend.
43. Take the sum of 4 m^ + 2 mn — r? and — 3 m^ — mn
-\-2n^ from the sum of 3 m^ -f 12 mn + 10 n^ and — 2 m^ —
11 mn — 11 n^
44. Subtract x^-Soc^ -\-0(^ -2 — 3 x from a; + a^-3 + 2aj*
— X' — a?, and arrange the result in ascending powers of x.
45.
46.
47. 48.
49.
50.
ac
am
X 2 ax
5ahx
am
he
m
ex —3 ex
3cdx
-2em
(a-b)c
(a — 1) m
(l-e)x (2a + 3c)a;
51.
52.
53. 54
55.
3mn
3aa; + 2
ax-\-by x-\-
y
3
ax-\- y
-2pn
ca;-l
ex -\-dy ex —
dy
x-cy
56. Take 2 6c — 3 ac? from 3 ac + 12 6c?.
57. From ax -{- 3 ey -\- ^ dz take hx — 2dy — az.
58. From 3 ax -\- 2 ay -\- 10 take 2x — y-\-l.
USE OF THE PARENTHESIS 13
USE OF THE PARENTHESIS
Exercise 4
Simplify: 1. a + (3 a + 2).
2. 4a + (2a-10).
3. 5a + (3a-2) + l.
4. 2a; + (4aj-2/) + (3a;-2)+y.
5. (5m + 7)4-(3m-2).
6. 2c + (3cH-4)-(c + 2).
7. 5m-(m4-l)-(2m-l).
8. 62/-(42/ + 32;)-(22/-42;).
9. 3a + [a + (4aH-3)].
10. 8a + [a-(2a-l)].
11. 5 m — J3mH-(2m-l)J.
12. 2a + {-3 + (2a-l)i.
13. lla-[4a-f (lOa-6);.
14. -3a + S2a-(-a + l)S.
15. -2a-[-a-S-aH-lS].
16. 13a + l-4a-(10a-[a-l])J.
17. a+[a — ;a— (a — 1)J].
18. a-[-a-5-a-(-a-l);].
19. 5 a - [2 a - (a + 1) - {3 a - (a + 1) - 1 S - 1].
20. 6 a - [(3 a + 1) 4- 53 a - 6 - (a + 2) - 3 aj - a].
21. (5a-l)-[5(3a-l)-10a + 5J-a]-J-(a-2)j.
22. (3 -f a - 1) - [- a + (a - 2 a - 3) - 3 a] - {a - 1 + aj
14 USE OF THE PARENTHESIS
23. (4a2-[3a2-(a4-2)-4]-|a2_^3a-(a-l)|)-a.
24. l-(-l) + (-l)-|l-[l-(l_lTa) + a]-aj.
25. l_J-[_(-l)-l]-li-(-J-[-(-l)]J).
26. Sx-\a-{2a-l3a-{5a-[7a-{Sa-x)J)J)\.
27. -m-[Sn-{-\2p-Sm-(m-n)l-{-7i-\Sm-{m-{-n)\^.
28. (4:a-a + 2)-\a-a-^{a-2)-2\-a-a-{-l.
Simplify and find numerical values of the following when
a= 5, 6 = 3, c = — 1, d = — 2, and x = 0:
29. a + 6 + c 4- d + a;.
30. 2a-\-c-3b-d.
31. a + 2c-(d + 2)+2(c + l).
32. a + c — d — 36 + a;.
33. 4 6 — d + a6 — aa;.
34. 2a-(c + d)-(a + d).
35. 4a6a;+(a — d) — (6 — c).
36. c +3a; — 2(a + 2a;)+4a.
37. a + 2 6 — [3a + c — S2a — (a; — c)|].
38. a + 3c — \b + 2x—(a — c)l-]-2d.
39. 3 a — [5 6 — (a6 + d) — a-d] — (a6 + a;).
40. a2-5a;-[3a6 + 462-(a6a; + d)].
41. aa;-[-64-(a-d)2 + a]-26.
42. (a + c)2+(a-c)2-[a6 + (2 6 + a;)2].
43. (a-\-c)x — (abx + a)^ — (d + c).
44. (a + a;) 6 - [ V2 a + c - (d + c)].
45. ic-[c+(a + d)2-26]2+(a-V^6^^)l
46. a62-S3(6 + c)2-[-c+2a(4d+6^]-[6ca;+V3a-c]j.
I
USE OF THE PARENTHESIS 16
Exercise 5
1. Insert the last two terms of a + b — c in a parenthesis
preceded by a plus sign.
2. Insert the last two terms of a — 6 — c in a parenthesis
preceded by a minus sign.
Insert the last three terms of each of the following expres-
sions in a parenthesis preceded by a minus sign :
S. a — b — c — d. e. a-\-b + c-{-d.
5. a-^b-c-d. 8. 4a*- a^-f- a^- 3a + l.
9. 6a^-a^-10a*-{-a^-a\
Collect the coefficients of a, of b, and of c, in each of the
following :
10. Sa — 2b-\-c — ma — nb — pc.
11. 5a — Smb-\-2nc — 2a — 6nb—3c — ma,
12. 10a — 4^b — 2c — ma — nb-{-pc.
13. —Sa— 4:C — pb — na — mc — b.
14. —c — a + 4:b — Sc-\-da — 10b,
15. —4:a — 3mb — 10c—b — ma — nb-\-pc^da.
Collect the coefficients of like terms in each of the following :
16. 4ar^-2ic + 3a^-3a; + 4a^-a^.
17. -Sx^-2x-\-x*-3x^ + 2x-x\
18. aa^ — ba^—cx — cx^ — dx + a^.
19. a^ + a^ — aa; — ca^ + 3 a^ — 6a;.
20. — aoi^ — cc^ — x — 4: x^ — ax^ — px — ca:^.
21. —px^ — cx-\-3xr — mx — nx^ — mx^ — abx + Sdx^ — cu?.
REVIEW
Exercise 6
Find the numerical value of :
1. (x-\-2y-{x-\-l)-4: when x = S.
2. (x-2)(x-j-5)-2{x + iy when x = 4:.
3. (a + 6)2 - 2 (a - 1) (6 - 1) - ab- when a = 5 and 6 = 3.
4. Vic^ + m^ — (n — myY when a; = 4, y = 0, m = 3, n = 2.
5. 2 a — [a — (3 a — 2 6)] when a = 3 and 6 = 2.
6. 3a— [4 6 + 2m — 3c + (a — 6) — 2a] when a = 4, 6 = 3,
c = — 2, and m = 1.
7. Subtract the sum of a^ + a^ — a and 2o?-\-2a — l from
the sum of 3 a^ — a^ — a + 1 and 2 a^ + 4 a — 3.
8. Take m^— 3m^ from the sum. of 2m*— m^ + 2 and
m^ — 3 m + 1.
9. From the sum of a^-{-a^ — a and a^ + 4 take the sum
of 4a3-2a2 + 17 and a - 3.
10. If ^. = 0^ + 0^4-1, -B = 2a;2-.T + 2, and (7=a^-3a; + 6,
find the value of ^ + J5 - C.
11. With values for A, B, and C equal to those in Ex-
ample 10, find the value oi A — B-\- C.
12. To what expression must you add 5 a^ to make ?
13. What expression added to a;^ + a; — 1 will make ?
14. — cc^ — a; + 3 is subtracted from 2 a;- — 3 a; + 4, and the
remainder is added to — 3a?2 + 2a; — 14. What expression
results ?
16
MULTIPLICATION 17
MULTIPLICATION
Exercise 7
Multiply :
1. a + 1 by a-f2. 6. a2 + a + 1 by a + L
2. a — 2 by a — 5. 7. a- + a + 1 by a — 1.
3. a + 4bya-3. 8. a'^+Sa^+Sa+l by a+L
4. a- -hi by a2_^2. 9. x'-2x^-{-l by a^ + L
5. a--{-3 by 2a--5. 10. 4 a;' + 4 a.- + 1 by 2a; + l.
11 . a- — 3 a — 2 by a- — a + 1.
12. a''-Sa'--\-3a-l hy cr--2a-\-l,
13. 12?/2-G?/-f-2 by /-32/-f-L
14. a4_4^.3_|_(5^^2_4^^^ j^ ^y a^-Sa^ + Sa-L
15. 5a^-2a2 + 3a-l by 3a2 + a-l.
16.* m^-2m''-h3m2-2m + l by m'^+2m^+Sm^-\-2m-^l.
17. m" - nr -\-Sm — 5 by m^ + m^ -f 3 m + 5.
18. a- + Z>--2a6-46-h4«-f-4 by a-6+2.
19. x^ + x'^ + l-\-x by 1 + a;2 4- a;.
20. l-Yx' + x-'-^Bx by _ 4a^+ 1 -f 2a;2.
21. l-4a'-4.a + Ga'-{-a' by Sa-l+a^-Sa^.
22. .t^-2.t2 + 3 by l-aj2^if.
23. or -\-2ab + h" -{- 7n" by a^ + 2a& + 6^ _ -^2^
24. a^-a^-^a-1 by 1 -f- a + a^ 4. ^3.
25. 5x'-2x' + x'-3 by - 3x' -^2x' -2 +x\
26. a^ H- 2 a5 -f ?>^ by ct" - 3 a-& + 3 ab'- - h\
27. a^ - 4 o&-^ + ^' + T) a7>2 _ 4 a-^& by 52 ^ a- - 2 a6.
R. & S. EX. IN ALG. — 2
18 DIVISION
28. 3a^b-2 a^V + 4 a6« by 2 a^h ^V- ah\
29. 5 a^m + 4 aW — 3 «%^ — am** by 2 a^m — 3 am^.
30. (a + 1) (a + 2) (a + 3).
;a + 2)(a-2)(aH-l).
a + 1) (a - l)(a + 3)(a - 3).
> + 2) (» + 2) (« - 2) (a; - 2).
V + m-\-l) {m^ — m -f 1) (m} — m
;3 a,.2 _ 5 ^ _l_ 3) (^ _ 4) (^ _ ^ ^ 2),
31.
32.
33.
34.
35.
36.
-1).
[a -\-h)-\-2 by (a + &) + 3.
38. (a + 2) + 3 a; by (a + 2) + 4 a;.
39. (a + c) — 4 by (a + c) + 6.
40. (a + 6) + (c + d) by (a + 6) +2(c + d).
Divide :
1. a^ + 4a by a.
2. 3a3 + 6a2 by 3a2.
3. a^ — a^-\-or by a^.
4. a"* — a^ + a- by —a^
DIVISION
Exercise 8
5. Sa + lOa^ + loa'^ by 5a.
6. 12m=^-9m2+3m by 3 m.
7. — ??i"^ 4- ?yi- + 4 m by —m.
8. 15a252_75a6 by -5 aft.
9. 27a^6^ + 36a26^-18a&^ by -9a&^
10. a^.^ 4a + 3 by a 4- 1.
11. l + 8a + 15a2 by l+3a.
12. a34-3a2 + 3a + l by a + 1.
DIVISION 19
13. Sa^-{-12a^-\-6a-{-l by 2a + l.
14. 16-32a + 24:a^-Sa^ + a' by 2-a.
15. a^-4a3 + 6a2-4a + l by a'-2aH-l.
16. m^ -\- 5 m^n -{- 5 mn^ -i- n^ by m- + 4m7i + 7t^.
17. 4m^-9m2 + 6m-l by 2m^-[-3m-l.
18. a^-10a* + 40a3-80a2 + 80a-32 by a2-4a + 4.
19. m*-3m3-36m2-71m-21 by m^-Sm-S.
20. 4a^-15a-^c + 26aV-23ac3 + 8c* by 4a2-7ac + 8c2.
21. m^-6m^ + 5m2-l by 7/1^ + 2 7/i2_^,t_l.
22. a^ + 12a2-48 + 52a-17a3 by a-2 + a2.
23. m'* 4- 4 mV + 16 71* by m^ + 2 mn -f 4 ?r.
24. 18c* + 82c2h-40-67c-45c3 by 5-4:C + Sc\
25. 4a;2-(-13a;-6a:3 + 6 + a^-2a;* by 1 -{-x' + 3x + Sx^.
26. 25 m - lOm^ + 15 + 14 m*- 41 m^ by -5m — S-{-7m^'
27. -14c*d + 12c^ + 10c3cZ2_c2d3_g^^^4_^4^^5 ^y
2cZ3-3cd2-4c2d + 6c^
28. m^ — n^—p^ — 2pn by m — n—p.
29. c^ + d^ + 77i^— 3 cdm by c + cZ 4- m.
30. a'^4-6^ + c3-3a6cby a^ + fts^c^-ac-ftc- a6.
31. x'-y^hj x-y. 38. 64 + 27 a*^ by 44- 3 a.
32. o^-fhj x-y. 39. 16-81 a* by 2 + 3 a.
33. a;*-2/*by a;-2/. 40. 125 0^-8 by 5 c- 2.
34. x*-y^hj x + y. 41. 216 771^-27 by 6 m-3.
35. «3-27bya;-3. 42. 81 a;*- 16/ by 3a; + 2 i/.
36. a;*-16by a; + 2. 43. 9m*-49 c« by 3 7n,2^7 c*.
37. 32-m^by2-m. 44. 27 d^ 64 6^ by 3 c^H 4 6'.
20 MULTIPLICATION BY INSPECTION
MULTIPLICATION BY INSPECTION
Exercise 9
The square of the sum of two quantities.
Expand by inspection :
1. (a + 6)2. 5. (a + 3)2. 9. (3 a + 0)1
2. {a + cy. 6. (a + 5)2. lo. (2a + 3a.f.
3. (a + m)l 7. (a + 10)1 11. (J ax-\-^y.
4. {a + 2y. 8. {2a^hy, 12. {^a? + dhy.
The square of the difference of two quantities.
Expand by inspection :
13. {a -by. 17. (a-2cy. 21. {2a-^cdy.
14. (a -my, 18. (3 a -2)2. 22. (3a2_2a5)2.
15. (a -4)2. 19. {db — cy, 23. {4.ax-xyy.
16. (a -6)2. 20. {a?-ahy. 24. (9a2-10c2)2,
The product of the sum and difference of two quantities.
Multiply by inspection :
25. (a + 6)(a-6). 30. (2 a6 + 1)(2 a& - 1).
26. (rt + c)(a-c). 31. (4a+3)(4a-3).
27. (a + 2)(a-2). 32. (pa^ -2c)(p 0^ + 2 c).
28. (a-4)(a + 4). 33. (S> a" -7 ah)(Qa? + 1 ah).
29. (a2-3)(a2 + 3). 34. (5a;^/-3a^)(5a;^/+3a;;^).
MULTIPLICATION BY INSPECTION 21
Perform the indicated multiplications:
35. {a-2c)K 41. (a6c-2)2.
36. (a-{-Sxy. 42. {arm -\- mn){am — mvi).
37. (a2-2)(a2 + 2). 43. (4 ac + 7)(4ac-7).
38. (a^-l)(a^ + l). 44. (6m2-3)l
39. (a3 + 3)2. 45. (8m-h5)(8m-5).
40. (a6 + 5c)2. 46. (3 (r' + 5)(3 c*- 5).
The product of the sum and difference of two quantities obtained
by grouping terms.
Multiply by inspection :
47. [(a + ?>)+c][(a+6)-c]. 55. (a'+a-irVjipi^-^-a-l),
48. [(a4-«)+3][(a+a;)-3]. 56. {a-x+y){a-x-y).
49. [(a-2)+c][(a-2)-c]. 57. (a-x-^y){a-irx+y).
50. [(a2+l)+a][(a2+l)-a]. 58. (c-d4-3)(c+d+3).
51. [cH-(a+6)][c— (a+6)]. 59. (a+m— n)(a— m+zi).
52. [m4-(w-p)][m-(n-i))]. 60. (c+cZ-3)(c-d+3). .
53. (a + 6 + c)(a + &-c). 61. (x2-l+a;)(ar^-l-ic).
54. (a+a;+2/)(a+a;— 2/). 62. (o^+a;— l)(a;2_^_j_j^)_
63. [(a + &) + (c + ^)][(a + &)-(c + cr)].
64. [(m + w) — (a;-2/)][(m + n) + (a;-y)].
65. (a — m + n — l)(aH-m+n4-l).
66. [m«-3m2-m-3][m3-|-3m2 + m-3].
22 MULTIPLICATION BY INSPECTION
The square of any polynomial.
Expand :
67. (a + 5 + c)2. 71. (a^ + a - 2)1
68. (a + 6-2c)2. 72. (2x^ -S xy - 5f)K
69. (a-\-b-c- df. 73. (a^ + 2 a^ - 2 a - 3)1
70. (a + 26 -3c + («)'. 74. (2a^-ar^2^+3a;2/'-22/3>)2^
The product of the forms {x ±a){x ±b).
75. (a; + 2)(a; + 3). 82. {x-^){x + 12).
76. (a; + 3)(a; + 5). ' 83. (a^- 3)(a^ + 7).
77. (a? + 10)(a; + 7). 84. (a:^ _|_ 4^)(^^ _ i3>)_
78. (a; + 3)(a; - 2). 85. (a^ + 2)(a.'3 + 19).
79. (a; + 4)(a;-5). 86. (y?f ^-l){x'y'' -2).
80. (a;-4)(a; + l). 87. (aa; + 13) (aa; - 12).
81. (a7 + 3)(a;-16). 88. {a'cx-ll){a^cx -\-Z).
The product of the forms {ax ± b)(cx ± d).
89. (2 a; + 1)(3 a; + 5). 95. (a' - 3 a)(4: a^ -\- 2 a).
90. (3 a; + 2)(4 x + 3). 96. (4 ac + 7 m)(3ac- 67/1).
91. (3a-7)(2a + 3). 97. (llx'-3y%5x' -\-f-).
92. (3m-^Sx){Sm-llx). 98. (mn - 13 1/) (5 mw + y).
93. (4a + 3c)(3a + 2c). 99. (llm^-37i){3m^-\-nn).
94. (2c2-7)(3c2-ll). 100. (6cd-3dm)(4:cd-\-Sclm),
DIVISION BY INSPECTION 23
DIVISION BY INSPECTION
Exercise 10
The difference of two squares.
Divide by inspection :
1. ^2- 62^ a- 6. 6. 25m2-81-^5m^-9.
2. a^-9-^a-\-3, 7. 81 ?>i^ - 49 -- 9 m^ - 7.
3. ci2_16--a-4. 8. 64 a^ - 36 ^ 8 « + 6.
4. 4a2_l^2a+l. 9. 169 a" -9 -r- 13 a + 3.
5. 16a2-9-j-4a-3. 10. 4m^ - 225 -^ 2m2- 15.
The difference of two cubes.
Divide by inspection :
11. m^-ji^-j-m-n. 17. 8 a« - 1 ^ 2 a - 1.
12. a^-b^-i-a-b. 18. 125 m^ - 27 -- 5 wi - 3.
13. m^-l-i-m-l. 19. 1 - 512 m«-^ 1 - 8 m^.
14. C3-8--C-2. 20. 343(^-d^^7c-d^
15. 2T-x^-^3-x. 21. 125-216?7i»-r-5-6m3.
16. l-64a3--l-4a. 22. 1000 - mht^ Sr 10 - mil.
The sum of two cubes.
Divide by inspection :
23. m^ -\- 71^ -i-m + n, 26. 64a3 + 27 ^4 m + 3.
24. 8 + C3-2 + C. 27. 125a3 + 8--5a + 2.
25. 27m3 + lH-3m4-l. 28. a^m^ + 27 ^ am + 3.
24 DIVISION BY INSPECTION
29. a«m« + 64 -r- aW + 4. 31. 216 a^ + 512-f- 6 a + 8.
30. 8a^-f343^2tt + 7. 32. 1000 a^ + 729 ^10 a + 9.
The sum or difference of like powers.
Divide by inspection :
33. a'^-b^-ha-b. 39. a^ + 32 -j- a + 2.
34. a^-b^^a-b. 40. 16a^-l-f-2a + l.
35. a^-b^-^a + b. 41. 32 a^ + 1-1-2 a + 1.
36. a^-\-b'-i-a-[-b. 42. 81 - 16 a* -J- 3 - 2 a.
37. a^ — ¥-^a — b. 43. 64 — m« -s- 2 + m.
38. a^-16H-a-2. 44. 32 + 243 a^^ 2 + 3 a.
Give the binomial divisors possible for each of the following :
45. a^-8. 48. a* -81. 51. 9a*-16. 54. a«-9.
46. a* -16. 49. a^-64. 52. 81 - a\ 55. a^-\-b^.
47. a^ + 32. 50. a^-b\ 53. 16 - a^ 56. a^^ _ 512
Give the quotients of the following :
57. a2-16--a + 4. 62. 125 a^ - 8 -^ 5 a- 2.
58. a^-16--a-2. 63. 100 c^ - a^ -f- 10 c^ + a^
59. 4a*-25--2a2+5. 64. 36 m*- 1 ^ 6 m^-f 1.
60. 81 a* - 36 c^ -f- 9 a^- 6 c. 65. 64a«- 27c^2^4a2-3c^
61. lOOm^-l-^lOm^-l. 66. 1 - 81 ««-^^ 1 + 3 al
67. 100(a4-l)'-9^10(a + l) + 3.
68. 27(a + l)^-8c-^--3(a + l)-2c.
PARENTHESIS WITH MULTIPLICATION 25
USE OF THE PARENTHESIS WITH
MULTIPLICATIOK
Exercise 11
Simplify :
1. a4-(a + l). 9. (a + 4)(a-2).
2. a+(2a-3). 10. 2a-\-S(a-3).
3. a + 2(a + l). 11. {a-{-l)(a + 2)-a\
4. a + 2(a-4). 12. (a + 2)^ + (a + 1)2.
5. a2+(a + l)'. 13. (a + 4)^ - (a + 2)2.
6. a + 2(a-l). 14. 7 a^ - 2 (a^ + 1).
7. (a+2)(a-l). 15. 2(a4- 1)'- a(a + l).
8. 5a + S(a + 2). 16. 4(a - l)2-3(a + 1)1
17. 4a + (a 4- 1) (a -1) -(«-!)'«
18. 2(a + l)(a4-2)-(a + l)(a-2).
19. (a + l)2-(a + l)(a-l)-2a.
20., 3 a^ + (a + 3) (a - 1) - (a - 3) (a + 1) - 2 (a^ - 2 a).
21. 2 (a + 1)' - 2 (a + 1) (a - 1) +3 (a - 2)\
22. (a + 3)2-2(a + l)2 + 3(a + 2)2.
23. (m + n)^ — m{m-\-nf — n (in + n)^.
24. m (m + n)2 -|- n (m — ?i)2 — (m + n)^
25. (a — m)(a — w) —(a — m)(a — p) — (m — a) (n — i?).
26. (m -{- n 4- p)^ — m (n + j9 — m) — n {m + p — n) .
27. (3m4-l)(3m-l) + [mn - Jl-m (2w-9m)J].
28. 3 a - [2 a + 3 (a - 1) - 2 (3 - 2 a)] - 4.
29. 2[3a-4(2a-l) -3(-2a-^r:^)].
30. 116-5[3-2 J8 + 3(4-2[8-3T^])|].
26 SIMPLE EQUATIONS
SIMPLE EQUATIONS
Exercise 12
Solve the following equations :
1. 3x + 4: = 2x + 5. 6. 2(x-\-S) = A + (x-2).
2. 5aj + 9 = 2a;+.15. 7. 5 (x-2) = 3 (a;+l)-l;i
Z. Sx-4: = x + 12. 8. 7aj-(;^j-3)-12 = 2a-.
4. 4a.' + 3 = a; + 6. 9. 3 (x + 2) +x^ = 5 -\- x\
5. 5a; + 7 = 2 0^4-9. 10. (a^+l) (a:+2) =a;(a;+l).
11. 4 + 5 (a; + 2) - 9 a; = (a; 4- 2)2 - x\
12. (x + 2)(x-5) = (x + 4) (x - 1).
13. (a;-l)(a;H-3)-2(a;4-l)(aJ-5)+a;2^0^
14. 2(a;2 + 2a; + l) - (a; + 2) =2a;2 + 6.
15. (x + 4:y + (x + iy= (a; + 3)2+2a;(a;+l)-a^.
16. 3 (a; 4- 5) (a; + 2) - (« + 3) (a; - 1) = 2 ar^ - (a; + 7).
17. (4a;-l)(a; + 3)-4a;2_(_i0^_^3)_^(3^Q ^
18. 2[a; + a^(ic-3)H-l]=(2a; + 5)(a;-l).
19. 5S2(a; + l) -(a; + 3)S =3[a; + 2Sa;-5(3-a;)S].
20. 2 [3 a^ 4- (aJ - 2) (a; - 1)] = 3 [2 a^ + (a; - 3)] + 2 a^.
21. 3[5a;-(a;4-3 4-2a;-l)] = 3a;-4 53a; + 2(a;-l4-3a;)J.
22. [(a;-2)(a;4-l) + (a; + 3)(x4-2)] = [(a^4-3)(aj-5)
^(x-5)(x + l)l
23 . (a; 4- 2) (2 aj 4- 1) (3 a; 4- 3) = (6 a; - 3) (a; 4- 1) (a? + 3).
24. aa; + a = 4a. 27. 2cx + d = 4:G^-\-d.
25. 2ax-{-c = 5c. 28. ax — (a-\-b) = 3a+b.
26. 4aa; — 5c = 5a — 5c. 29. 2(a — a;)=8a.
PROBLEMS IN SIMPLE EQUATIONS 27
30. 3(x-{-a)-\-2{x — a) = 6(a — x).
31. (a + b)x-^(a — b)x = a^b,
32. 5 a -\- (a -^ bx) c = ac — bcx.
33. 10(a + b) -\-3x=a-\-b — 5x.
34. S(a + b)x — 2(a — b)x = a-\-5b.
35. (x — m)(x — n) = {x — m — iif.
PROBLEMS IN SIMPLE EQUATIONS
Exercise 13
1. What number is that which, when doubled, equals 24 ?
2. What is the number that, increased by 12, equals 27 ?
3. If a certain number is increased by 12, twice the sum
will be 28. What is the number?
4. Four times a certain number when diminished by 6 is
equal to 12 more than the number. What is the number ?
5. There are two numbers whose sum is 77, and the greater
is 13 more than the smaller. Find them.
6. A man is 13 years older than his brother, and the sum
of their ages is 49 years. Find the age of each.
7. A father is 4 times as old as his son, and the sum of
their ages is 55 years. Find the age of each.
8. The sum of the ages of three brothers is 85 years. The
oldest is twice the age of the youngest and 5 years older than
the second. Find the age of each.
9. A child is 3 years older than his brother, and 5 times
his age is 3 years more than 6 times his brother's age. Find
the age of each.
28 PROBLEMS IN SIMPLE EQUATIONS
10. Five years ago a man was 4 times as old as his son, but
now he is only 3 times as old. Find the present age of each.
11. A man bought the same number each of 2-cent, 5-cent,
and 6-cent stamps, paying ^ 0.91 for the lot. How many of
each kind did he buy ?
12. Find three consecutive numbers whose sum is 39.
13. Find three consecutive odd numbers whose sum is 33.
14. Find two consecutive even numbers, the difference of
whose squares is 52.
15. A man bought a number of horses at $ 150 each, twice
as many cows at ^40 each, and 3 times as many sheep at
$ 5 each. The lot cost $ 1225. How many of each kind did
he buy ?
16. How can you pay a bill of $ 80 so as to use the same
number each of 1-dollar, 5-dollar, and 10-dollar bills ?
17. A man asked a farmer how many cows he had, and was
answered, '' If you gave me 18 more, I should then have twice
as many as I now have." How many had he ?
18. A man sold 15 hens, receiving 80 cents each for a part
and 50 cents each for the remainder. He got ^ 9.60 for all.
How many were sold at each price ?
19. Three dollars in nickels and dimes were distributed
among 42 boys, and each boy received one coin. How many
boys received dimes ?
20. Into what two amounts must $ 1700 be divided so that
the income of one part at 5 per cent interest shall be double
the income of the other part at 6 per cent interest ?
REVIEW
£zercise 14
1. If a = 3, 6 = 1, c = 0, and cl = l, find the value of
a — (a — b) + \a — {b + c)l — [a — (b — c — d)].
2. From what expression must you subtract the sum of
5 a^ 4- 8, 3 a 4- 2 a^, and a^ -\-a—l, to produce the expression
4a2-8a + 3?
3. If a = 7n-\-n — 2p, b = m — 2n-{-p, and c = ?i+p — 2m,
show that a -j- 6 4- c = 0.
4. What must be added to a^ + a^ — 2 a 4- 3 that the sum
may be — a^ — a^ 4- 2 a — 3.
5. To what expression must x* — 3x^ + 2a^ — x-\-5 be
added to produce a^ — x — 1?
6. What is the numerical value of the remainder when
3a4-2c — d is subtracted from 4a4-3c — 2d if a-\-c = d
and d = 7?
7. If A = a^-l-^4:0^, B = -x-2x'-\-l, and C=2a^-i-
2 0)2 + a; + 1, find the value of - A-[B- (2 A- C)-\- C^
8. Simplify
4a- [- 6c- (- 54-26-3 (^) -4 a] -5- (4c -3 cQ.
9. Simplify
i-[-i+!-i-(-i+?-i+(-i)s-i)n-
10. What is the coefficient of x in the reduced form of the
expression (x — 4a) — [2a — Six — 2 (x — a)j]?
29
30 BEVIEW
11. Multiply m^ — 2m» + 2m- — 2m + l by m* + 2m^ +
3 m^ -{-2m-\- 1.
12. Multiply a*b - a^b' -\- a^b^ - ab^ by a^b - d'b^ -\- ab^
13. Multiply ^a^ — I tt2 + a — 1 by i a — 1.
14. Multiply 0.1 aj2 + . 04 0^4-0.5 by 0.1 x-^ + 4 ic + .05.
15. Divide Gm^ + ^/i^ — 29m2 + 27m — 9 by 3m^ + 5wr —
7 m + 3.
16. Divide 1.2 aj^ - 2.9 a^ + -9 a;^ + a; by .3 a; -.5.
17. Multiply by inspection (a -{-b — 2)((i—b — 2).
18. Multiply by inspection (a -\- b — c) (a — b -{• c).
19. Expand by inspection (a — 2b-\-Sc — d)\
20. Divide 1 by 1 — 3 m to 5 terms.
21. Divide m'^ by m + 2 to 4 terms.
22. Simplify (5 a + 1) (a - 3) - (2 a - 3/ _ (a - 5) (a + 3).
23. Simplify (2a-l){a-{-4:)-2a'-\3a-{-(2 a-l){a-6)l.
24. Simplify 5 a + (4 a - 1) + « + 3 (« + 1) - (a + 3) (a + 1).
25. Simplify 4 a^ - 3 a [a^ + a^ - (a - 2)] - 3 (a + 1) (a - 7).
26. Find the value of (a + bf — (a + c)^— 2 (a + 6 + c) when
a = l, b = 2, and c = 0.
27. Find value of -^ + V6^-4ac ^ ^j^^^ ^^5^ 5^_11^
and c — Z.
28. Find value of -?>-V&^-4ac ^ ^^^^^^ ^^^^ & = -ll,
and c = — 3.
29. [(a; + a)2 4- 5(a; + a) + 4] -- [(a? + «) + !] = ?
30. [5(a; + m)4-3][5(a;4-m)-3] = ?
31. Solve 2a — 3a;(a + c) = 5a + 3c.
FACTORING 31
32. Solve x'-(x-\- af = (a + If.
33. Solve a'-\-{x-l){x-2) = x' + (a-l){a + l).
34. Solve 4(a; + 4)(a;-3)-2(a^-2) = 3(a; + l)(x-4)-cc2.
FACTORING
Ezercise 15
Factor :
1. a^^a^ + a. 10. Ga-^-Qa^ + Sa.
2. m^ — m^H-m. 11. 5 m — 10 7)i^ + 15 m\
3. 2m + 4m2 + 6m». 12. 12 m' - 18 m^ + 24 m«.
4. 5c2+10cH-15. 13. 5ac— 10 6c + 5 cd.
5. a24-4a^ + 6a*. 14. 4 a^c _ 10 ftc^ + 6 ac.
6. 3a^ + 9aj^-6a^. 15. 6 a^y -\- S xy^ - 9 x^.
7. 10 a.-3 - 12 ar^ + 13 £c. 16. m^ -S m' + 4.m^-m^
8. 8a:^-12x»-16ar2. 17. a^c - aV + aV + ac^
9. 4a3-f 8a2 + 12a. 18. S a'-\-^a^-2 a^ -{-6 a^
19. a'b-a%^-{-a^b^-ab\
20. 4a*6-12a^62-16a^63 + 8a*6^
21 . 15 ay + 150 ay - 225 a/ + 15 a/.
22 . 48 7n^n^ - 144 mhi^ - 192 m«ri^ + 240 m'^'n^K
Exercise 16
Factor :
1. a2 + 4a + 4. 4. a^-20a; + 100.
2. a2 + 6a + 9. 5. a2-18a + 81.
3. a2-8a4-16. 6. 4m2H-4m + l.
32 FACTORING
7. 0,2 _^ 22 a + 121. 18. 25 a' - SO a -}- 9.
8. 16c2 + 8c + l. 19. 9a2-30a + 25.
9. a2-36a + 324. 20. 49c2-84c + 36.
10. 36a2-12a + l. 21. 16 a^b^ + S abc -^ c".
11. 9-6a + a2. 22. a^ + lSa^ + Sla.
12. m2 + 42m + 441. 23. 36 a^ + 60 a'c -\- 25 a^c^.
13. a-^>--14a6 + 49. 24. 144 m^ - 240 m/i + 100 w^.
14. ify + 32a'2/ + 256. 25. 121 x^ - 374 a.-^ + 289.
15. cv'cH^- 10 acd-\- 25. 26. 625 m^ - 50 ?7i + 1.
16. 64-16m?z + mV. 27. (a + 6)2 + 2(a + 5) +1.
17. 4a2+12aH-9. 28. (a + by -^6(a + b) + 9.
29. (a-c)2-6(a-c)H-9.
30. 25(a-m)2-70(a-m)+49.
Exercise 17
Factor :
1. a'-b\ 11. 64a;2-25. 21. 81a^-49a.
2. a2-4. 12. a*-l. 22. 81mV-16.
3. a2-4m2. 13. a^-Slc^. 23. 324- 256 a^/.
4. c2-9fZ2. 14. aV-25. 24. 289 - 16 m^.
5. 0^2-16. 15. 9a*-4a«. 25. (a + 6)^-1.
6. m2-49. 16. 6a^-24a. 26. (m-ri)2-4.
7. 9 0^-16. 17. 3 ar^- 75 X. 27. 7ri'-(n-^py.
8. 25 a^b^ -9. 18. a^6^-81. 28. 9a''-(b-cy.
9. 36c2-25. 19. 121 a^- 49. 29. 4(a + &)2-c2.
10. 36c2d2_9^ 20. 64a*-a8. 30. 16(x-yy-9.
FACTORING 33
31. 4a2_9(a + l)2. 34. 36 (a + &)' - 49 (m - w)^.
32. 9 a^- 16 (a + 2)2. 35. {a? - Wf - 4. {o? -{- Wf
33. 2^{a-hf-{c-\-df. 36. 100 - (a -f- 6 + c)^.
37. 81(a + 6)2-4(a + 6 + l)2.
38. 9(a2 + & + c)2-16(a2_&_c)2.
Exercise 18
Factor :
1. a^ + 2ah-\-h'^-c\ 11. (? -^ cd-\-^dr -lQ>m\
2. m2-2mn + n2-p2^ 12. 4«6-4 a^-f- 1 - 61
3. x'^ — h'^-[-y'^ — 2xy. 13. in^ — A mn - 9 7n-n^ + 4: n\
4. m2 + ?i2-j92^2mn. 14. 9 a^-2522 + 16/ + 24 a-?/.
5. m2+n2-^2_2y^n. 15. 4a2 + 12a6-9c2 + 9&l
6. 2m + n2-m2-l. 16. 20 mn -^ p^ - 4: m^ - 25 n\
7. l_a^-2a;y-2/^. 17. 4a2 + a*- 4a3-l.
8. ar^_4a2-4a-l. 18. 5 ar'-S-Sa^-lOa^.
9. 2mn + mV + l— p2. 19. 8ac — 4 a2_4c2 + 4.
10. ci--c2-l-2c. 20. m^ + 16n^-16-\-Smn.
21. a--^2ab + b- — 7nr — 2 mn — n\
22. 4a2-4a + l-9a.'2-|-6a;2/-/.
23. 9a'-30a + 25-4.b'-4:b-l.
24. a2-c2 + 62_^2_2a&_2cd
25. a%^ ^ 10 a'mhj - n^ - 1 _ 2 n + 25 /.
26. 711^ — n'^ — x^-\-if — 2 (my — wic).
27. 25 a^ + 1 - 1 6 a^ _ 9 c^ - 10 a2 _ 24 a^c.
R. & 8. EX. IN ALG, — 3
34
FACTORING
28. 5a2 + 562_5m2-10(a6 + m7i)-5w«.
29. -12a6 + 2 + 24a25_i862_^18a262_8a*.
30. 3a2 + 1262_l2a4_i47 52_34^2^_-|_2«5.
Factor :
1. m^ + m^n^ + n*.
2. x^-lx^y'^-\-y\
3. iB*-5a^3/2^4^^
4. m*-23m2 + l.
5. a^-79a2 + l.
6. m*-171m2 + l.
Exercise 19
7. 25a^ + 66a262 + 496^
8. 49 X*- 11 a;22^2_^ 25 2/^
9. 16a;*-73a^ + 36.
10. 49a^-74a262^25 6*.
11. 289m^-42mV + 169w^
12. 16a*-145a262 + 9 6*.
Factor :
1. a2 + 3a + 2.
2. a2-a-12.
3. a2-9a + 20.
4. ar' + 5a;-24.
5. a2 + 18a + 17.
6. c2-llc + 24.
7. m2-19m + 88.
8. c2-9c-22.
9. a;2^5a._14.
10. a2-3a-28.
11. iB2^9a.^i4^
Exercise 20
12. 2/'-ll2/ + 28.
13. ar^-9a; + 14/
14. c2 + 42c--43.
15. m2-4m-165.
16. 2/' + 12 2/ -108.
17. aV-21aa;-46.
18. a262 + 13a6+40.
19. aW + 21 am -130.
20. c2d2 + 9cd-52.
21. mV-2??i2n-35.
22. ic2«2_20fl;;2-69.
FACTORING 36
23. x^f-xy-12. 34. m* — m*-156m^
24. a;^-13ic2 + 36. 36. a^+ (a + &)a; + a6.
25. ic^ — 9 a^ — 22 ic. 36. a;^ + (m + n) a; + mw.
26. a;* — Sar' — 9. 37. a^ -\- {c + d) a -{- cd.
27. a^ — 7a^ — 78 a. 38. x^ — {m-\-n)x-{-mn.
28. a262_6a6-187. 39. a;^^ (a4- 26)a; + 2a6.
29. 16-6a-a2. 40. a^+ (3a + 2 6)a;+.6a6.
30. 18-19c+c2. 41. iB2_(a-6)a;-a6.
31. 147 - 46 a--^ - a;«. 42. x^-ax-hx + ah.
32. 90m24-13m^-m«. 43. a^- (3m-2)a- 6m.
33. a:3_^i0a:2_963.^ 44^ a^-\-{m-2m7?)ay-2m?ii?y\
Ibcercise 21
Factor :
1. 2a2 + 5a + 3. 14. 8a2-30a-8.
2. 6ar'-a;-2. 15. 24m2-14m-49.
3. 2ar^-3a;-9. 16. 2 a;^ -f 7 a; - 15.
4. 2a2 + 7« + 3. 17. 18a2-f-9a-2.
5. 8a^+^a;-3. 18. 40 ar^ - 61 a; + 7.
6. 6m2+^m-5. 19. 8m2 + 2m-3.
7. 15ar^ + lla; + 2. 20. ^b a" - IS a - 12,
8. 7ar^-41a;-6. 21. 6a2 + 25a-9.
9. 6a2-29a + 28. 22. 8m2 + 5m-3.
10. 3a2-19a + 6. 23. 4.2 x" - 11 x - 20.
11. 12c2 + 17c-5. 24. 16m2-67n-27.
12. 6/-y-12. 25. 12y^-y-20.
13. 3m2-llm + 6. 26. 2x'-4.x-12^,
36 FACTORING
27. 12 m^ - 7 m^n- 12 m?il 36. 20 a^^^ - 9 a^ft^ - 20 a&.
28. 6m3 + 29m2-22m. 37. 16 0^^2 + 2 ccZ- 3.
29. 8x^-26^^2 + 18. 38. 75 a - 210 aa; + 147 ax^
30. aW-9aW + 20a2m. 39. 48 a;^ - 176 x* + 65 aj^.
31. 26a34-197a2 + l5a. 40. 55x-«-x^-2x3
32. 52a2-153a-52. 41. 2(a + l)2 + 3(a4-l) + 1.
33. 3m2-30m + 63. 42. 3(a + 1)^- 8(a + 1) + 4.
34. 12x^-25x2 + 12. 43. 2(a + l)2 + 5(a + 1) + 2.
35. x«-x*-42x3. 44. 2(a-2)2-5(a-2)-3.
45. 3 (m - 1)2 - 11 (m - 1) ri + 6 nl
46. 10 (a + 6)2 - 11 (a2_ 62) + 3 (a -6)2.
Exercise 22
Factor :
1. m^-n\ 8. a365-8. 15. 343 «%« - 729 7i«.
2. m3-27. 9. 27- 8 c^c^^^ 16. {m^nf + 3?.
3. 8a3 + l. 10. 7MV-343. 17. (c + d)»-8.
4. 27-8c3. 11. 64a3 + 125. 18. (a + 1)^ + 64.
5. 27a3 + 8. 12. 8a«-27. 19. 27-8(a + 2)3.
6. 64- 125 c^. 13. 64a» + c^ 20. 64 (a + 6)^ - 27 a^
7. 125a3+2763. 14. 125mi2_^i5^ gl. (a + 1)^ - 8 (a + 2)^.
22. a^-h\ 26. a}''-2^. 30. (a -6)^-1.
23. a^-16. 27. 4m^-81. 31. (a-6)«-8.
24. 81 -m^ 28. a^^'-w}''. 32. (a + 6)^ -256.
25. a«-64. 29. m^^-'n}^ 33. (2a+l)^-16(2a-l)^
FACTORING B7
34. m^ + 8. 38. x^'^-^y^^ 42. x^^ + 1.
35. m^ + 1. 39. x^ — y^ 43. m« + 27.
36. a^ — l. 40. a;^ + 2/^ 44. 64a« + l.
37. 3^ -{-32, 41. i»8-^. 45. a}^ + 7n^.
Exercise 23
Factor :
1. m^ + mn + mp -{- n2). 13. m^ + 5m2 + 2m + 10.
2. ab-\-a-\-7b-\-T. 14. m^w —p^mn -f- mic — p^a;.
3. 2c7n-3dm-\-2c-Sd. 15. a^- 1 + 2(a2- 1).
4. ay-ab-bx-\-xy, 16. a^ (a^ - 9) - a (a + S)^.
5. a:3_^^_^a.^l^ 17. 5(a^ + 8)-15(ic + 2).
6. xy-2y-x^-{-2x. 18. a;^4-2a^-8a;- 16.
7. a^-a^ + a-l. 19. 2a2(a + 3) -Sa^-Sa + S.
8. m^-n^-m-n. 20. c^-\-4:C^-S.
9. a^-a^-a + l. 21. m^-lOm-SO.
10. 6a3 + 4:a2_9a-6. 22. 4a3-39a + 45.
11. m^ + m^-m-l. 23. a3 + 9a2 + ll a_21.
12. 15 ax-2llay+9bx-12 by. 24. a'^ + 3a2+ 3a + 2.
25 . am -{- an -\- ap -{- bm -\-bn-]- bp.
26. m^ {n — x) -{- m{n — X) —2 (n — x),
27. m^ + ?/^ + m — 2m2/ — 2/-^6.
28. (a-l)(a + l) + (a;-l)(a-l).
29. (a + l)(aj + 2)-(a + l)(y + 2).
30. a^-l-xia-l).
31. m^ + l — ^>(m + l).
32. (m + l)(m2-4)-(m + l)(m + 2)-m-2.
REVIEW
Exercise 24
Factor :
1. 6a2-fl9a + 10. 18. 24: a^b^ - 36 b* - SO ab\
2. 2x'-6xy-U0y\ 19. 7a;^-7a;.
3. a*-\-a. 20. m^~m'^-2m^
4. c*-^(^d^ + d', 21. m* + n*-23mV.
5. m^ — m^ — SOm. 22. 42/ + a;2 — 1 — 4^2
6. c* + cc?8. 23. (2m-5n)2-(m-27i)2.
7. a^ + 2xy-ix^-y\ 24. 4a%2_4ci2^4^8^3^3_3^^6^
8. a;4_iB3a + iK2;3_^2^^ 25. 25a^b^c'-9,
9. a^-oc^y + xy^-y\ 26. a^-Sa-a^ + S.
10. 1 — m^ — 2 mTi — n^. 27. a;^ — /.
11. c^ + (^4_-|^8g2^2^ 28. m^-rn?n-\-mn^-n\
12. m2wy_^2p_^?p_^j 29. 81m«-16m7i^
13. 6(?-c{d-V)-{d-Vf, 30. 72aH5a2_i2a.
14. a2-c2-4c-4. 31. l + aa;-(c2 4-ac)aj2.
15. d2 + 3cf-d^-3ci. 32. 54-16mW
16. 8a^4-27a;2^ 33. 4.^f-(^^y^-zy.
17. JK« + 2/3, 34. 49a^ + 34a^2^H-25y.
35. (a; — m) (2/ — n) — (a; — n) (y — m).
36. 15m'-14m^-8m3.
38
REVIEW 39
37. (m-n)(2a2-2a6) + (n-m)(2a6-262).
38. (a-l)(a-2)(a-3)-(a-l) + (a-l)(ci-2).
39. (2c2 + 3d2)a + (2a2 + 3c2)d.
40. d^ + d^-d-^l, 62. 2m3n-16ni^
41. x^-64:. 63. 64a;^-a;.
42. 1000 + 27 c«. 64. (a^+a -l)2-(a3-a-l)l
43. 5a3-20a2-300a. 65. a;^-27ar^ + l.
44. mn— pr+j97i — mr. 66. Slx'^—y^z^^.
45. c2-2cd + d2-l. 67. 5cd-12d' + 2c'.
46. 4-9(a;-32/)'. 68. 72 ic^ _|. gg ^ _ 40 ^.^
47. a'i»^-a^-a^-2ax. 69. 3iB*4-192a;/.
48. ISf-^Sy^-lSy. 70. 3c3-12c3d2_4^2_^l^
49. m* — 2mw«-n* + 2m»w. 71. pY-277^.
50. a;^ + 125a;/. 72. mV-a^^-m^ + l.
51. m3 + m2w + 2mn2 + 2?i3. 73. aj^ - 25 (a; - 3)2.
52. 24a^-5a;-36. 74. a^-c^
53. 72a^ + ^x — 4:5. 75. 5m^ — 5mhi~5mn — 5m.
54. (a2-6y-(a2-a6)2. 76. m^-27m2 + 162.
55. ac2 + 7ac-30a. 77. 9 a;^ + 68 a.-^ - 32.
56. x^y^ — a^ — 2/^ + 1. 78. 1 — m^n^ — ph-^ + 2 mnpr.
57. a;* + 4a^-8a;-32. 79. m^-5m2 + 4.
58. 5cV + 35ca^-90a^. 80. m^ - m^x - m -\- x.
59. ac + cd-ab-bd. 81. 24 a.-^ + 43 a;^ - 56 a;.
60. a^ — m^ -f- a; — m. 82. m*—(m — 6y.
61. 9 a;^- 66*2 + 25. 83. Sa}^ + a7n}'.
40 //. C. F, AND L. C. M.
84. 24:€rd'-A7cd-75. 87. a^c + 3ac'-3a^-(^.
85. 16a^-/-9 + 6y. 88. c^ - 64 c^ + 64 a^ - a^.
86. 12a3 + 69a2 + 45a. 89. a^ _ a^ft^ _ 52 _ ^^
90. 2a;3^3^_3^^3
91. 10 (??i + c)2 + 7 a(m + c)- 6 a^.
92. 100 + 10a;^-25x«-ic2.
93. (m+py-l-2(m+p-\-l).
94. a2 + 2a-c2 + 4c-3.
95. (2m-3)2-6(2m-3)?i-7w2.
' 96. x'6-13a^4-12.
97. m2 + 7i2_(l + 2mn).
98. (c-2d)^-9-3(c-2d + 3).
99. 6c3_25c2 + 8c-16.
100. x^y + y^z-{- xz^ — x^z — xif- — 2/2^.
HIGHEST COMxMON FACTOR AND LOWEST
COMMON MULTIPLE
Rzercise 25
By factoring find the H. C. F. and L. C. M. of:
1. a^-fe^ a^-h\ o>-h\
2. a^-ab', 2a*-2a'b% a^-2a^b + ab\
3. a^-16, a'-a-2, a2-4a + 4.
4. m2-3m + 2, m2-m-2, m2 + m-6.
5. «2 + a;-12, cc2-4a; + 3, ar^ + 2a;-15.
H. C. F, AND L, C. M. 41
6. o?-^2o?-l^a, a?-6a?-\-Qa, o?-2a^-Sa.
7. x^'-^x' + l, 2it? + 2x'-2x, a^H-2a^-l.
8. a;4 + a^_6, 3a^H-6a^-24, aj^-lOx^ + lG.
9. 2x'-nx-^0, 3a^-25a; + 8, x'-x-m.
10. 5m3-5n^ 15(m-7i)^ 10 m^ - 20 m?i + 10 nl
11. m* — n*, m^ + m?n — mn^ — n^j m* — 2mV-f n*.
12. 5a3 + 406^ 7 a« + 28 a^d + 28 aft^ ^ a' -12 a?h\
13. c^-d^ c^ + d^ c^ + d^^ (? + 2cd-\-d\
14. 12a^-30a;-18, 27 a^ - 90 a; + 27, 15 ar' - 42 a; - 9.
15. mn — mp + 2n — 2p, m^H-6m^4-12m + 8.
16. a;'' H- a^2/^ 4- 2/^, 3?z -\- x^v -\- xyz -{- ocyv -{■ yh -\- yH.
17. 12(m«-n«), 18(m*-n^), 24 (m^ - mn + n'^).
18. a^-h^-.o?h^ab, a? -\-W -a?h -al)", a" - ab^ + b* - a^b.
19. a^-3a'-^a-\-12, a'-lSa' + Se, a^ + ^a^- 9a -^18.
20. a^ + 3a^-\-3a + 2, a'-Sa-S, a^ -{-Sa^ + a-2.
21. a3-22a + 15, a3 + 6a2_25, a^ + 13a2 + 36a-20.
f
Exercise 26
Find the H. C. F. and the L. C. M. of;
1. a:3 + 4a;2 + 7a; + 6, a^ + 4ar^ + a;-6.
2. a:^ + 6x^-\-llx + 12, x^ ^2a^ -6x-\-S.
3. 2a;*-a^ + a:^-a;-l, 2x^ + Sa^ -x^-^x + 1.
4. 3a;^4-3a^-3a;-3, 4a;*-4a^-8a^ + 4a; + 4.
5. 6ar' + 19ar^ + 19a; + 6, 4. x^ -{- S a^ -\- 5 x -\- 3.
6. a;^ + 3«3 + 5a^4-4a; + 2, a;^ + 3a.-34- 6a^ + 5a; + 3.
7. 2m*-{-m^-9m'-\-S7n-2, 2m^- Tm^H- 11 m^- 8m + 2.
42 FRACTIONS
8. 2a* + 5a3 + 2a2_a-2, 6a« + 3a^ + 6a»-3.
9. 3a3 + 14a2-5a-56, 6 o? + 10 a" + 11 a -\-^^.
10. 4m* + 3m3-6m2-29m + 30, A m* - 7n^ - IS m^
+ 14 m - 5.
11. 4a« + 14a^4-20a3 + 70a2, 6 a« + 21 a^-12a4-42a3.
12. a^-^a^b-\-6a%^-ah^-6b\ a^ -S a^b-\-3 al/ -2b\
13. c^-2c3-7c2-|-8c-10, c^ + c^-9c2 + 10c-8.
14. 2 m'' — m^n — 11 mV + 17 mw^ — 7 n'^, m* — 2 m^/i — m^n^
+ 4 mn^ — 2n\
15. a;'^4-2a;--10ic-21, «34.4a^_2a;-15, a^4-2cc2_7 3._-{^2.
16. 2aXa'-2a^-7a'-\-16a + 7), 5 a\a^- 5 a'- 23 a -S),
6a(a3-6a2-26a-9).
FRACTIONS
I. TRANSFORMATIONS
Rsercise 27
Reduce to lowest terms :
V a^ + 27 ^ r2a^-x-6 ^ a;^-81
0^2-9 12i»2-13a; + 3 aj^+lSic^+Sl
Ax' + ^x + l 3?^ -6 16 - (m + ny
5a;^4-5 ' ' 6m^-24* * {m-Af-n'
2a^- 2a -12 a44.ft3^^2_^q_|_]^
• a'j^2d'-a-2' ' a'-l
^ y? + m.r 4- »-« + 'nin -- 7j^^— 64 ^
a;2-h3mx + 2m2 * * m^+2m3-8m-16*
4m^— 8mnH-4n^ 12 m^ — m^^^ 4- m^n — ti^
FRACTIONS 43
14.
15.
16.
6 ac — 2 ad — 3 ?)c + hd
9 ac — 3 ad H- 3 5c - hd
(m 4- ^)^ + 7 m + 7 n + 10
w?-\-2mn + 'n?- 4(m + n) — 12*
,^ 10a^ + 5a;^-105a:3 ^, a^4_7 ar^4.12 a; + 4
24 a;^
-64a;3-
-24iB2
m'-
in^ — ^m
+ 3
2'
m^ — m —
1
:^-
lla; + 6
18. """^ » 22.
8m-3
19. ^^ ' Jl ' 23.
a^_l_2aj2_7a;_2
x3_20a>+33 a;*-14ar^ + l
2-5a-4a^+3a^ 2^ 5m^-5mV
4-|.4^+9a2+4a«-5a** ' 2m«-2m2w + 2mn2*
Exercise 28
Change tci mixed expressions :
x-\-2 ' ' x^-^x-S
B2_12a;-47 ^ 4 a^+12 a;3_^^4
2.
a; + 3 2a;3 + 7
a^_2a^4-4a;-l ^ fl^-a^ + 3a; + 2
iC-
-3
3a3
a — b
9.
m^-\-2
10.
m^ + m — 1
x'-\-3
x'-^x-l ' x'-{-2x + 2
12
2 + a;-a^
7. .^. 9. „ ^ .. 11. "^
19 1
12. — =— to 4 terms.
44 FRACTIONS
Exercise 29
Change to improper fractions :
, ^ x — 1 e 2 2 I in^nHm + n)
2. a — Sc-\ — • 6. (c-\-d)^—^ ^*
a + c c-\-d
4. 1
a^ + ft^
•• ^ ^ ' ^ a + 1
(a-^by
8. ^^(^' + 3) 2.
9. m^ —
2
mn
+2
m
- 71^) 4- m V
2m?2+7i^— »^ ,„ -, , / 9 I I 1 \
10. m-{-n—p — ^- 13. l + (ar + a;H •
m + wH-p \ x^lj
!b2+3x-2 L V ^ /-
12. a+2-^+6^+a'. 15. (''-^-A+^ + 2.
a + 4 V=" + l /
ie.l-[.-{2..^-^i)i--.)].
17. 3m-r("^ + 2X'^-^)-(-»^ + ^)1-(«t + 2).
m + 3 ^ ^
18. a'+r-'»6-a'+(-a' + «6)-&n I „ ^,
a — 6
19. (a-6)»-r(a + 6) + 3«H&-a)+a(«'-a)-n
Collect :
FRACTIONS 45
11. ADDITION AND SUBTRACTION
Exercise 30
1. 5^4-3^ + ?. 12. ? + -^+ ^
3 4 2 a a+1 a-1
2 4a;4-l . 2a; — 1 ^^^ __a ^ _^ 1
4 3 a-1 a + 1 a'-l
2a; + l a?-l 14 3 5 2 m- 3
' 3a;2 2a;3 ' * m m-1 m^-l*
,1.1 ,,^ 2 3.1
a; (a; 4-1) a; (a; — 1) 3 — 3 a- 5 — 5a l + a
2 , 1 1 ^^m — n.n—p.p — m
5. -H -— r -• lb. 1 i
a; — 1 a; + l ar — 1 mn np mp
g + l ' g- 1 --, 4a — 6 18 a6 . 4a+ft
* a-1 a + l' ' a+2 6 a2-4 62 a-2 6'
7. a + 2 a-2 18. "^ "^ ^ ^
a-2 a-\-2 a^-1 a;-l
„ 5a + l 2a + l ,« c^ + c?^ c^ d^
o. — -— • ly.
aH-3 a-3 cd cd + (P cd + c^
9. ,^±4_ . ^^. 20. -^-^^+-^ + . 1
(a-c)2 a'-c' a^^x-\-l x-1 sc^-l
6
^Q (m + ny _ (m-n)\ ^^ 1 1__^
m — n m-\-n a^+2x-{-4: x—2 oc^—S
11. -A_ + ^_ + ^. 22. 2^2m _^ m+l 1
m + 1 m — 1 m — 2 (m— 1)^ (m— 1)^ m—1
46 FRACTIONS
23 a+r. a-m ^^_ a^^_a_^
ar-{-am-\-7rr ar—am+mr a -^2 a — 2
25. -^ + 2--i-. 28. 3c--^ + -i-.
x+1 x—1 c+1 c— 1
30. (ra+-J^\-U+-J^\ 32. 2-^+^+^
\^ m—nj \ m+nj \^—y ^+y) x^—y^
33^ K + ny _/m_^7i_^^
mn{m — nf \n m
31. ^±1_24-^^-
a;4-2 a;-4
34 a? — 3a , 9aa; , 1
ic2_3aa; + 9a2 a^ + 27a3 a;4-3a
35. 4J^ + ^ + J^. 36. ^+ 3 1
a^-1 1-a a + 1 2-a a-\-2 a^-4:
3^ g + l 4 g-l
a + 3 9-a2 3-tt*
38. 1 + -J^+ 2 2 3
X a; — 1 l—a^ x-\-l x-\-7?
39. _A___1__^+ -8
3m4-15 125-5m=^ 7m-35
40 ^ — 5 I ^ + 5 . 21m
5 + ^1 5 — m m^ — 25
., 1 , 20a , 2 1
a l-lGa'' 8a-2 a + 4a2
42.
1 +6-^ +
FRACTIONS
1
47
a^3 ' S-a ' 9-a2
2 Sx' 4
43. 3x +
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
x-S x-\-S 9-«2
m + 1 1 m^ H- m — 2
m^ + m + l m — 1 1 — m*
c — a? c + <? _ • (d — c)^
c + c? c — a; {c — x)(c — d)
x — Sa
3ax~2y? 1
ar^_3aa; + 9ci2 a^ + 27a« 3a + a;
a+1 , a+4
(a-4)(2-c) (a-l)(2-c)
(m — n) (m — p) (n — p) (n — m) (p — m)(p — n)
x"
+
2/^
+
(a; -2/) (a; -2;) (y-z){y-x) (z-x)(z^y)
m + 1
m-1 2
+
m^ — m + 1 m* + m^ + l m^ + m + 1
-1— f-i L_l
_^ [o 1 ( 1 , ^ \\
x + b \ x-n \a?-25 x-5)]
\2x + 2-(^ l_^1_-2^_ (1_^__A_2^
-J— f-J— (^ ( 1 +^-M1
1-x [x-2 \&-x-i? \2-Zx + ii? x-X)]_
48 FRACTIONS
III. MULTIPLICATION AND DIVISION
Exercise 31
Simplify :
1.
m^ — n^ m^
w? (m — Tif
^2-144 a + 3
a2_9 ^_^i2
w? — n^ m^ — mn-{- n^
m-{-n
m^ + n^ m—n
7nf-\- mn + n^
(c 4- df ^ c'-cd + dj'
c3 + (Z3 * c2-f-2cd + c«'*
m2-2m + l
(m 4- 2) (m - 1) (m + 3) (m - 2)
' m-\-n \n m)
mn /m + ^^ j
7n-{-n\. n m
— >
c-d ' c^-d^' (? + df
10. (^^/-^j(^^2(^2/2_^i) + i)-
a^-{- {m +p)a + mp a^ — n^
12. fg-Sm + mAfl+i + i-T— — 4^— — l-
FRACTIONS
49
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
3 (m — yi) m^— {p — 5)m — 5p ' m-\-p
m(m^-\-5m) m^ -{- mp — mn — np m—p
a^ + ac + ab + he ^ g? + «/+ «cg + df
(J? -\- ac -^ ad -^ d/: c? -\- od -\- ab -^hd
m-\-n .m
A _^ fm-^n _ m — rb \
i) ' \m — n m-\-nJ
+ d c-(A /6 3 S y
— d c + (ZyVc c — d c-\-d)
m — n m-\-n^
'c + d c —
c
m^-9 ^ (m-4)^ . |^ m^-7m + 12 7/i + 4^
* m + 3,
6 a2-2a-3
a^-g
+ d c — d\
— d c-^dj
a^^a-2 a^-2a + l
c2 + d2 1^-
4 m^ — n^ —p^ — 2np_^ 2m — n—p
A 7n^ + n^ — p^ -^ 4: mn 2m-\-n-{-p
a^-5a;-f 6 ^ a^4-a;-2 . a;^-4a; + 3
a^^Sx + 2' x' + x-e ' arH-4a; + 3*
(-•D(S-')H'-?>
3 cd(d -c) + (^-d^ r dF\ . c^(c-cg)^
(a_cy-&2 (g + & + c)' (g + cf - b^
(5 + c)2-a2* la-i-by-c"' a^-(b-^cy'
Tin} — 2 w^v? -\-n^ m^ — r^ , in
m^ + ri" tti^-mH^^-n'' ' (m^ + w«) (m^ + n^)
4
R. & S. EX. IK ALG
x{x — 1) + 1
50
FRACTIONS
27. f£_Ji-^-2Yl 2^Y
\yz xz xy xj\ x + y + zj
28. f™-i2iY^r('-+— -4Y-+— +4\
\n mj [\n m J\n m j
29 (c + d)^-4 d'-{c-2)^ . c^d-2
' 4 - (c - d)2 (^-{d-\-2f c-d-2
"■ [e-'-'-?)-e-'-^)]*[('-^)fe)}
[_\xy xz yzj xy J
Simplify :
1
1.
— X
+ 1
lY. COMPLEX FRACTIONS
Exercise 32
a4-2
4.
a-3 "^
16 ^
a?
8 10_3
X* a^ sc^
m + 2
2.
3.
m
m-3 + -
m
a + l
4
a;
a^+l
1
1
ajH-1
x-\-l —
5.
a; + 4 —
« +
a; + 2 .
X —
x-^1
8.
d' c"
i + i+1
m n p
n p 771
FRACTIONS 51
x^ — x — 6
10. -o — 71 ?r- 11
12.
13.
a^-x-6
• a.'2-6a;-7
c—d c+d
c+c? c — d
c'-d' ^
ic-df
2 3
a; -14
a & a 6
16. 1-
4a
^ 2m , , _J__\
Vl+J- 2m + V
18.
2m
2
1 — a;
(f-')-(f-')
I4.I i_l 20. a + =— .
a h a h a-\-
1 4 . • 1
& , a h a a + -
a
ft^ + a-2 3fa4.2)-2(a + l)
a2_2a-3 .
15- 2 ■ Q^_i_9 ~^' 21.
tt2_4a4-3
10 10 2^-
a?
a3 + 64
a^-
.4a« + 16a2
m + 1
m + 1
1
5 5 m + l
v2
^^' a? , g^c + Z * ^ + ^-^
^z-c+(^:r^2 100^+9^-63
52
FRACTIONS
24. 2a +
3a +
4a-l
26.
1 +
1 +
l-x
25.
1 +
4 wn
(m-2nV
4?nri
1_^
27.
1
1
1
a-?>
aH-6
b'
—
a^
264-1
a — b
28.
29.
m^ — 71^ (711 -\- 7i)(m — p)
m? + 7i^ m^ — mp — mn 4- np
\n'' m?i my \7r mn my
l+3a l-3a
l-3a l+3a
3a 1
3a
l+3a l-3a
30.
np
mp
1
mn
(m-
-P)^-
-ii^
1^_
np
1
mp
1
mn
31.
a/*^ + mx + nx-\- mn
x^-\-nx — px — np
a? + mx — nx — mn
x^—nx -\-px — 7ip
32.
14- c3
1-
1-f
8c3
1-c
1 +
2c
2c
14-2C
33.
rx + yV ^ /x-yy
x-y V / 'x + y V
34.
m4-n ■ m^ + n^ '
m — n
m'
m^ 4- ^i^
+ n
m'
FRACTIONS
53
35.
36.
(m — n) (m + p) (m -f- n) (m — p)
{m — n)(m -\- p) {m + n){m — p)
37.
3a; — 1 X
3 "^4
^-l(x+^^-)-\-2x
38.
1 _ ^ + ^ ' I _ c^ + ( ?"
39.
2mn
4:mn
m^ — 4 wm + n^
-r- 14 m?i .
\ mn J
40.
i_i i_jL,i
' ah a"-
a b
X
+
11
62 a6"^a2
62 -a^
a'b^
41.
r46(a + 26) J (a3-86V 1
|_ ft-26 JLa^-8q262_^l()6^J
a2 4- 2 ty6 + 4 6^
a + 26
(m^ -\- n^)(m — n)
m^ — mn + 71^ m^ — 2 mr?
(m — ny m^ H- mw + n'
54
FB ACTIONS
43.
44.
- (»--)f'+-y
^
1+
a
45. Find the value of 1 when a = x-\-l.
a
a^ — b^
46. Find the value of — when a = ic + 1 and b = x — l.
47. Find the value of (m-iy-(m^-l) ^hen m = a + l.
(1 — my
48. Find the value of
a-\-m , a — m Sam _. „^ _ 3a
1 when 7n = — •
a — m a-\-m cr — iiw 2
49. When m = - and n = ^ find value of ^>^' + mn - 2 7i^ ^
50. Ifa = ^^and6 = ^^,findvalueof ^i^ + 1 ^~^
iC + l
x-\
a — h
a + h
x-\-
51. If a; = a — 2, find value of
x^l
-1
a; +
+ 1
52. If a = m and 6 = w, show that
x-1
a + h a — h 4 mn
a — h a + h w? — r^
53. If a = -T-^ and c = . "^^ , find value of a in terms of m
1— c 1— m
and reduce to simplest form.
64. If a; = -^ and y = -^; show that ^"^
2a6
a-f-6
a — h
y? + f (x^+^
FRACTIONAL EQUATIONS 56
FRACTIONAL EQUATIONS
I. NUMERICAL
Exercise 33
Solve:
2 a; a; a; 1
8. ^--l + 5-^- + 2.
3 5 2 6
3 2
2 ^_? = ^ + ?. « 4a; + l , 2a;-l 3^ + 3
* 3 3 5^5 ^' '~S~^~~~2~~'~2~'
3. 1 + 1 = ^-1. -^ 5a; + l , 3a;-2 _ l + 8a; .
5 3 15 10. —^--^-^ — ^— .
* 3 2~4 3* „ a;4-l a;-l _ 3-a;
^ 2 a; , a; a; 11 /^
6 ^+l4.^±^ = 2.
• 2^4
11
12. 2(a; + 3) 3(a; + l) ^o^
5 2
13. i.(a; + 2) = i(a;-3).
2a; + 3 a;-l_o
11 ^ 3 14. ^(a;-l)-|(a;4-l) = 0.
15. |(a; + l)-f(aJ + 2)=i(a; + l).
16. (a; + l)(a;-i) = a^. .
17. |(2a;H-l) + 2 = ^(3a;-2).
18. ^-i(. + 3) = ^-3.
19. a;-J3a;-i(a;-h 1)1=0.
20. i[a;-(2a; + i)] = i(a; + i).
56 FRACTIONAL EQUATIONS
23. 2__ 3
24. fe^V(£=31^=.(5.^-l).
25. 2i-|(a^ + 3) = l|^+l-2^.
o o
26 fe±D-i(^±ll = 31 ^^'-1 ^ 3a; + l
3 2 ' * 4a; + 3 6a; + l
27 -A_ = _3_. 32 3a;4-2 l^o;-!
ic + l x-{-2 ' 4.x 4 ic + 1*
28. _i- + 2 = ^^. 33. 3a^-l _ 2a;-3
a; + l x-1 ^x + 2 Qx-\-5
29. ^±1 = ^^. 34. _L-+-i- = l ^
a; + l a; + l x-l x^-1
30. ?^-l = ^Zl3. 35. ^_+ 3 0.
ic + 2 a;-2 x-2 x + 2 3^-4
36. 3_^— A_ + i= 9a-
3a; + l l-3aj 9iB2_-L
37. ~i- = -J^ ^ 38. ? + — ^L_ =
a-'^ + l x+1 x'-x + l ar^ + 3a;H-2 (a; + 2)2
39 a;^ + a;4-l ^ a:^ — a;4-l . x
x + 1 x-1 "^l-ar^* .
40.
FRACTIONAL EQUATIONS 67
X 5 a?^ X
3a;4-6 6^^-24. 2ic-4
41. „ ,^ ,+ 2 1
2a;2 + a;-l a!2-x-2 2ar'-5a! + 2
43. ^ l- = ^i 1-.
a; + 2 x-\-3 .t + 4 a;-f-5
44. -1,+ 2 3
45.
46.
47.
48.
x-\-l a^ — 1 1—x
2 03*^ 3 a; x
a^ — 1 1 — x x-{-l
3 4.2
2a; + l 4a;2_-i^ l-2x
xj-3___2_^a^--l
a;_2 ic2_4 2 + a;*
3 2 5
x-S l-2a; 2af-7a;4-3
4a; + l ^ 3 _ 1 +4a; ^
x-^2 ~a^-4 2-a; *
a;+5 5 — X 25 — a^
51. ^ 4 3_^o
2(2a; + 5) 2a;= + 9a;+10 3a; + 6
52. -^- + 5^ = ^-+1.
x-32(a?'-»-6) a; + 2
58 FRACTIONAL EQUATIONS
53. ^ + ^^ = ^_
6a;-6 3-3a; 2a; + 2*
64. 2a;-l 4a;-l .^ ^ -18ar»
3a; + 4 6a;-l 21a;-4 + 18ic2
55. 2 _ 3__^ . 1
a; + 3 2(aj + 3) 3(a; + 3) 4
_ 4a^-2a ;4-l . 4a^ + 2a; + l .
2x-l ^ 2x^1
^^ Sx-2 , 2x-l X
58.
6 2iB + 3 2
3aj + 5 2a;-l 2ic
18 x-\-5 12
59 a^ 4a; + 5 _ 2a; — 1
3 2a; + 4~ 6
60 ^^ + 13 6a;-1 ^ 3a; + 2
8 3 a;- 9 12 '
6 9 a; - 36 6
62. ^+^i- ^-2 -^J-l
63.
64.
66.
9
3 a; + 11
9
li
a;-l
a;-3
X -
-4 a;
-2
a;-2
a;-4
X -
-3 X
-1
2a;-3
3a;-
1
x-^1
2
4
5
4
1
5
3
3
2 a; -3
3a;-
J._
a; 4-1
1
= —
^BACnonAL EQUATIONS 6^
II. LITERAL
Exercise 34
Solve:
1. 3a; + 5a = ic + 8a. 6. ^ax = h{x-\-a).
2. 2a'4-4a = 3a; + 3a. 7. 2(x^a)=^{x-a).
3. 3x + 2a = 4a; + a. 8. (x + a)^ = (a; — a)^ + 4 a.
4. 3aa; + 4a = aa; + 10a. 9. (a; — a)^ = (a; — 6/ + al
5. 3aa; = a(a; + a). 10. {a — h)x + {a + h)x = a?.
11. (a + 6)a; + (a — &)ic = «^ + ca;.
12. (aj + a) (a; — m) = (a; — a) (a; 4- m).
13. mx (x + m) + 7ia; (a; — m) = (m + ?i) ar' + w — n.
14. a6 — (a — 2 6)a;=(2a — 6)a; — 1.
15. (x + 2a+by-(x-h2ay = b(Sa-^5b).
16. ^ + ^ = 3.
a 2a
17 ^_?^ = 1. ^
* a 3a 2
18. 'i^' + !^ = m^ + n^
19. i + l = l_l.
m a? 71 aj
20. ca; + a + - = --
c a
2\a J S\a
1 .
22.
x-\-m 3
a; — 71 4
23.
4 3
m + a; tti — a.
24.
aj + d^ g
a; + c2 d
25.
771 + 1 7/1 + a;
771 — 1 m — X
26.
n(a — x) t\
m — -^ ^ — a = 0.
2a-a;
OT
9 5
m4-7i — a; 77i — Te + a;
60 FRACTIONAL EQUATIONS
28. nk+Jl^^LlzJl. 30.
X — 1 X -\-l
'" ^ 31.
««7.
m m-
-n m-\-n
30 a; + n _
a; + 2m
33.
2a;-a
X — a
34.
2x-a
x+b
m-\-x
0? — 6
35.
2
m — x
2
1^
36.
1-^
1+^
1+^
m
1-5
m
m
37.
m-f 1
m — 1
2 ~
' 4
m + a;
a?
m — x
m^ — x^
ab -{-X
^
ab — X y? — a^l^
3m
3a; + 4mH-2n
x-{-l
41. t^Lzl^ = 4^Hhl^.
i«H-faJ -ia-fa;
42. a; + a _. a^ + « + l .
ic — a a; — a — 2
43. ^ — 2'^ m + 4a; ^^p
m+a; m — x
. x + a _ x 4- c
a; + c a; + a
45. E±^_^LL^ = o.
a; — 5 a — 6
c+2aj 4a^
46.
c — 2a; 4a;^ — c^
x
m
m — n . m
47. 1= —
OQ ^x — a _o b — x m-\ ^— ^
5a;-a ^ a-10a; 43. ^ !L_ = '?Lz:
• 2x-a a-4a;* * a;-m a;-n x
.f. X . a nx ._ 2a;— m x-\-2m , 5m ^
40. 1 = • 49. ' 1 = U.
m + n n — m w^-v? 2x-\-m x—2m x
SIMULTANEOUS EQUATIONS 61
SIMULTANEOUS EQUATIONS
I. NUMERICAL
Exercise 35
Solve:
1. 5x-{- y = ll,
5.
x + 3y = 5,
9.
5x — y==16,
3x + 2y = l.
3x-{-4:y = 0.
x = y.
2. x-{-2y = S,
6.
x-4.y = 7,
10.
a; + 82/ = -20,
3x- y = 3.
4.x- y = 13.
3 aj + 4 2/ = 0.
3. 2x-\-3y = 16,
7.
2x-13y = l,
11.
aJ-2^=0,
3x + 2y = U.
3x-21y = <d.
4a;-5y-h2=0.
4. x-\-2y = 3,
8.
2x + 3y = 4,
12.
4a;- 2/ = 10,
3x~ y = 16.
6x- y = l.
72/-2x = 12.
13. y — x — l =
3,
15. 5
x-3
:2/-72 = 5y,
x—5=-y
X
-1 =
= 15y.
14. 3x = -2y,
16. 5
x + 3
2/ = 102,
x = 35-{-lly.
7
2/ + 3
ia; = 104.
- M=^'
19.
1-!=^'
21.
x^y
2 3'
M=^-
| + . = 9.
i+i-»-
- M=-^'
20.
X y_7
3 4 4'
22.
hh-^
X y 5 .
3 2~ 6*
5 3 5
-
3 2 9
62 SIMULTANEOUS EQUATIONS
''■l+t'' -1-1=^' -T-H'
3 4 4 2^3 2 3 ~ ^'
26. ^ + f = -3A, 32. .-^ = 2 + 2(.-,),
33.
2x y
3 3
1
3'
14 ^
a;
= 7'
32/ a^
5 8
-k-
2a;4-3
5y-S
-2,
7
4
i(142/-
ox 160^-
23
* 2
3 -^'
3a; + l
.2/ + l_i
4
3
a; y
.3(0^ + 2/4-1)
3 2~
4
!-«=
= 1 + 12,
aj+2/
5
35-2^
a;
a;
"3*
x + Sy
3
-2 42/-
— x + 5
5' '
3a;-2/
+ 7 2a;
+ 32/
' + 1
10
12
5
2
34.
35. -,
4a;+82/+l 7a:-22/-l
2x+4:y 5x-^6y Ay
se. ^-^t.=(2.-|)
<?o 6 2/-1 , 3a; + l _5
30. — - — + — ^ — -^,
3^+1^14
5a;-l 2/ + 3 , I^a -22/ 5*
4 7 "^3 '
a;+2 y-2 Sjy^Ax) ^^- x-y'^'
^^' ~S 2- = — 4 '
y X X y
4 2 5 3 _41
2 ^3 ^^^ -' 3 5 ~60'
^_5=^-3(2/-a.)
SIMULTANEOUS EQUATIONS 63
38. -^_=_-^_,
4a;-3(a;-?/) _^ ., x-2(x-\-y)
39. (a; + 3)(2/-l) = (aj-3)(2/ + l),
a^-^ + i^ = (.-3)(, + l) + llf
.^ 4aj-3 . 2x-3 Qx-l
40.
2
r
Sx-2y
" 3 '
52/-
1
x + 1
15^-10
3
3x-y
9
2x
3
Sy
5
x-h2y
4
=3--7
6y
>
1+^
,_S
\x-y
5
-^+t
41.
2a5-2/ + 3 05-2^ + 3
3a;-4y + 3 _ 4a;-2y — 9
4 ~^ 3 '
43.
(K 2/
45.
5+5=2,
a; y
47.
X y
a; y
a; y
i-l=o.
a; 2/
44.
13 5
25 2/ 2'
46.
3 2 31
a; 2/ 40'
48.
1 1 _ 5
2x^Sy 12'
? + l- = -2.
a; y
5_10_11^
X y S'
111
3a; 2y 12*
64 SIMULTANEOUS EQUATIONS
49. ^ + ^=1, 51. A + X = _2, 53. 1^-A = _A,
2x Sy ' 2x 3y ' 3x 2y 36'
X 2/6 4:X Qy ' ■ 8x62/ 24
50. A + A = _??, 52. ?_5 = _4, 54. Ah_A = 8,
3a; 22/ 3 a; 2/ 2a; 42/ '
2a; 32/ 3* x y ^* 3x 2y 3*
4a; dy 3a; 22/
^ ?_ = 22 -62/ + 4a; = 26a;2/.
3a; 32/ ^'
3a;-5 , y-1 /3a;-l\ , ^
— 2 — + 3" ^^-y- 2/f— 3 — \ = xy-x-\-6.
II. LITERAL
Exercise 36
Solve:
1. a; + 32/ = 7a, 5. x-\-y = m,
5a; — 22/ = 18 a. 2a;— 32/ = w.
2. 3x-\-2y = 5a, 6. a; + 2/ = '^ + ^>
5a;-f32/ = 8a. 3a; — 22/ = m — ?j. ^
3. aa; + 62/ = l> 7. x — y = m — n,
ax — 6y = 3. nx-\- my = 2 mn.
4. 2aa; + 3&?/=l, 8. ax-\-hy —ntj
3 ax-]- 2 by = 2. ex -{- dy = n
SIMULTANEOUS EQUATIONS 65
9. x-{-my = -l, ^^ (7n + n)x-\-(m-n)y ^^
y = n{x-\-l). ' m'^ + 7i^
10. mx = ny,
x-\-y = a.
11. {c-{-d)x=(c — d)y,
x — a = y.
x — y = 0.
13. (711 + n) X -{- cy = 1,
ex -\- (m-\- n) 2/ = 1.
14. ^ + ^ = c,
a; 2/
X y
X y _ 3cd — c^
' c-\-d c — d c^ — d^'
x + y = c.
16. i^ + - = a + &,
ox ay
X y
^ • y ^ ^ 25. (g + ^>-(^-^)y -i^
ma; — ny = m'-^ + w^-
20.
(a + 6)x+(a-6)2
21.
a c
x-\-l 7/H-l ^
n — J-
c a
22.
aU-^a-,-^^
ah
23.
x-\-y , x-y ^
i»-?/ « + ?/_n
m n
24.
2' + »-« = 3a.
17. -^ + -
4ccZ
X . ^ ^ 2 (£zi^' = l..
+
m — n m-\-n m^ — n^ (c + c?)2/
18 ^+^ ^ ^-^« 26. ^-^ I y-'^ =i
' y + a y—2a ' p—m p — 7i
x — a __ x-\-Sa x + m . y— m _ m
y—a y+a ' p m—n p
R. & S. EX. IN ALG. — 5
66 SIMULTANEOUS EQUATIONS
III. THREE OR MORE UNKNOWN QUANTITIES
Zizercise 37
Solve :
1. Zx-\- y—2z = l, 9. x-\-y = 2a,
2x-3y-{- z = -l, x + z=3a,
4:X—2y-\-3z = 14:. , ^
^ y + z =2a.
2. x + 3y+ 2 = 1,
2x+ y-3z = l, ^^' ^ + 2y+ z = a,
3x-^2y-2z = -2. x+ y-^2z = b,
3. 2x + 3y-5z = 0, 2/+ z + 2x = c.
3x-4:y-2z = -3,
2y-3x + Sz = 7. '^' o^ + 20 = 2(, - .),
4. 2. + 3, + 4. = 12, y + 20 = 3iz-x)^
3a.-42/ + 5. = 2, . + 20 = 2(.. - 2,).
4:X + 5y-\-6z==24.. 12. x + y-^z = a-\-h,
5. 3a;- y-{-2z = -ll, x-^y-z = a-b,
3y + 2x- . = -12, 2/ + ^-^ = c-a.
32!+ x + 2y = -20.
6. 2a;4-2/-10;2 = 20, 13 ^4.^ + ^ = ^
^ , Q 1K 3^2^4 12*
— y — oz-\-3x = lbj
^ 4^3 2 12'
7. x + y = z + 3y
X y . z _ 5
y = Sx-8, 2~4 + 3 2
2 — a? = 4.
8. x + 2y = 25, ^^' 1^; + J2/ + J. = 23,
2/-22; = 0, icc + iy + i2 = 28,
a; + 30 = 2O. ix-\-iy + ^z = 27.
SIMULTANEOUS EQUATIONS 67
''• 2 + 2/ 5'
S+z f
z 2
4 + a; 3'
16. ay + hx = l,
ex + az= 1,
bz + cy=l.
"• A+ft-^
X
x + y-\-u = SAf
2/ + z + w = 36.
20.
1.2 1_^
X y z
2_4 + 3^_3_
X y z
?_l+?=i
X y z 2'
21.
_ + - = w,
a? 2/
a; z
2, z
2|,
2* 32/ 2
1+1+ i=r. 22. i5+§_§=4,
X y 2z X y z
9 4 4.
18. a; + 2/ + 2; = 33, -"7, + ^"=^'
X y z
+ z+u = S5, .
? +5-1 = 2.
352/ -
.o 1^1 ^ 23. ^^ = 7i,
19. _ + _ = -, x — y
X y Q
1_^1_7 -^==li
^ + ^"12'
1 + 1 = 1 -^^=^i-
a; 2
4 2/-^
68 SIMULTANEOUS EQUATIONS
24. " + -^ + 5 = 1
X y z 2
25.
mx + ny = a(m-{- n),
mx -\-az = n{a-\- m),
a . b c 1
26.
ny -\-az = m(a + n).
x + y=22,
^_^_£— _
1
2/ + ^ = 18,
X y z
2
•
2; -j- W = 14,
W-\-U = 10f
w + a? = 16.
IV. PROBLEMS IN SIMULTANEOUS EQUATIONS
Exercise 38
1. A man purchased 20 acres of land for $1640. Part of
it was bought for $ 90 an acre and the remainder for $ 50 an
acre. How many acres were there in each portion ?
2. A man and a boy together weigh 230 pounds, and twice
the man's weight is 60 pounds more than 3 times the boy's
weight. Find the weight of each.
3. Three horses and 4 cows can be bought for $610, but
at the same rates it takes $720 to purchase 4 horses and 3
cows. Find the price of each per head.
4. If half of A's money is added to B's money, the sum is
$ 170 ; but if half of B's is added to A's money, the sum is
$ 160. How much money has each ?
5. In 10 hours A walks 1 mile more than B walks in 8
hours. In 5 hours B walks 5^ miles less than A walks in 7
hours. How many miles does each walk per hour ?
6. If the numerator of a certain fraction is subtracted from
the denominator, the remainder is 21 ; but if the denominator
is subtracted from 8 times the numerator, the remainder is — 7^
Find the fraction,
SIMULTANEOUS EQUATIONS 69
7. In a certain town meeting 312 voters were present, and
a motion was carried by a majority of 8 votes. How many
voted for and against the motion ?
8. Two men had together $ 100, and if the first had given
$ 10 to the second, each would then have had the same amount.
How much had each originally ?
9. If 2 is added to both numerator and denominator of a
certain fraction, the resulting fraction is |. If 1 is subtracted
from both numerator and denominator, the new fraction is ^.
Find the original fraction.
10. A boatman can row 20 miles down a stream and back in
10 hours, the current being uniform. He can row 2 miles up-
stream in the same time that he can row 3 miles downstream.
Find his rate per hour both down and upstream.
11. If the width of a field were increased 1 rod and the
length 2 rods, the area would be 34 square rods greater ; but if
the width were decreased 2 rods and the length increased 3
rods, the area would be unchanged. Find the length and width
of the field.
12. Find a fraction such that if you double the numerator
and add 3 to the denominator the result is f ; but if you add 3
to the numerator and double the denominator the result is |^.
13. Two sheep-owners met. A said, " Sell me 4 of your
sheep, and I shall have twice as many as you have." B said,
"No; sell me 1 of yours, and we shall each have the same
number." How many had each ?
14. When a certain number of two digits is doubled and in-
creased by 4, the result is the same as if the digits had been
reversed and this number decreased by 22. The number is 2
less than 3 times the sum of its digits. Find the number.
70 SIMULTANEOUS EQUATIONS
15. If I divide a certain number by 3 more than the sum of
its two digits, I get a quotient of 3 and a remainder of 8. But
if I reverse the order of the digits and divide by twice the
sum of the digits in the same reversed order, my quotient is
3 and remainder 11. Mnd the number.
16. A boy bought 5 apples and 3 oranges for 25 cents, 4
oranges and 5 pears for 35 cents, 2 pears and 7 apples for 20
cents. Find the prices paid for each apple, orange, and pear.
17. Find 3 numbers such that if each be added to | the sum
of the others, the results will be 32, 28, and 30 respectively.
18. The sum of the three digits of a number is 12. The
hundreds' digit is one half the sum of the other two, and the
units' digit is ^ the number composed of the other two in
the original order. Find the number.
19. Three boys together weigh 300 pounds. Half the sum
of the weights of the first and the third equals the weight of
the second. The sum of the weights of the second and the
third divided by the difference between the weights of the third
and the first gives a quotient of 5 and a remainder of 20.
Find the weight of each.
20. A and B together can do a certain piece of work in 3
days, A and C the same work in 4 days, B and C the same
work in 6 days. How long will it take each alone to do the
work ? How long will it take all working together ?
21. Some books were divided among 3 boys, so that the first
had 12 less than half of all, the second 1 less than half the
remainder, and the third 17. Find the number each received.
22. A boy has 100 pieces of silver. The value of the quar-
ters is 3 times the value of the dimes, and the sum of the
values of the half dollars and dimes divided by the difference
of the values of the quarters and half dollars is f. Find the
number of dimes, quarters, and half dollars.
INVOLUTION AND EVOLUTION 71
INVOLUTION AND EVOLUTION
L MONOMIALS
Exercise 39
Write the value of :
1. (2a)l 9. {-2mn)\ ^^ (Sa^\ ^1. p^^V.
2. (3 ay. 10. (-2 ay. ^ ^ ^
^- ^^'*^'- 12. (^m'ny)\ ^g. ^-^^^Y. 23. ^^^'^'^*
/ 2 am V
V 3cdy
®- ^~-^")- 13. aaO^
7. (Say. ' \S
20. -f-^X 25. r^^^Y.
40.
^S2a''
8. (-2a2)^ 15. (-^aby.
26. V4m^ 34. -y/lQ m*n\
27. Vl6mV. 35. ^Wm^n"^.
28. ^8?d^. 3e. ^/4^. ^1- \'^K^J
29. ^27?.
30. V64mV.
^
42. A^/-32'''''^''"
31. ^327^. ^^ ^/25^ 43. V-27(a + 6)^.
32. VlOOa^y. 44. ^Sl (m-ny.
33. ■v/-64a^c^ ' >'343ci2 45. V-32(a + l)'
72 INVOLUTION AND EVOLUTION
n. INVOLUTION — BINOMIALS
Exercise 40
Expand :
1.
(a + by.
12.
(d^-
-4 c)*.
20.
(-
-!)•
2.
(a + by.
13.
(d^-
-3 c)'.
3.
(a + m)^
14.
(c^-\-2cy.
21.
(aH
-a^-l)^
4.
5.
(a + 2y
(a-sy
15.
22.
i^-
3 a; + 2/.
6.
(2a + 3y.
16.
(2 a
-I)'-
23.
(m*-
-7l2_3^^\
7.
iSa-2y,
17.
fab \'
24.
K +
■ a + 1)^
8.
(2a2 + 5)^
25.
(«=-
■a + lf-
9.
(ab - ly
18.
V2"'
-')■
26.
(a +
c-i)'-
10.
11.
(a-2y.
(a'b'-{-2cy.
19.
(i-
-11-
27.
(2(1-
-a'-\-iy
HI. EVOLUTION — SQUARE ROOT
Exercise 41
Extract the square root of :
1. aj* + 2a^ + 5aj2-t-4aj + 4.
2. x*-6x--{-17x'-24:X-\-16.
3. aj« + 2ar^ + aj'-2a^-2a;2 + l.
4. a;^_2a;3 + lla^-10a; + 25.
5. a^ + 4:a;*-8ar' + 4ar^-16a; + 16.
6. 4a;*-20x3_,_37 3^_3Q^_,_9
INVOLUTION AND EVOLUTION 73
7. ^-12x-2x'-\-4.s?-\-Q^.
8. a^ — 2 a^x + 5 aV — 6 aV + 6 aV — 4 aa^ + a;«.
9. 12a;3-30a; + 4a;^4-25-llar^.
10. 25 a;2y _^ 2 a;y H- x« - 8 ary- 12 a^?/^ + 36^^- 4 ar*?/.
11. ar'-2ar^-14a; + 49 + 14x^4-a^.
12. - + X^+ 3 +T+9
13 ^'_i^4.??^ + a; + i. .
^^' 9 3^6 ^"^^16
14. ^ + i^+2-i^ + 4
15. 4 + 4a-a^ + -+ — -- + -.
16. f + 4 + 6_i_2a.
9 Of a 3
17. c«-| + ^*-f-^' + <^ + f^-- + Ti-,-
2 16 2a 4aa4a^
Extract, to three terms, the square root of :
18. 1 4- 3a. 20. a^ + 9 &.
19. l-5a. 21. a^+4a;.
Extract the fourth root of :
22. 81a:4-216a^ + 216a^-96a; + 16.
23. a^-12a362_^54a26*-108a6« + 8168.
24. 16a^ + 16a»a; + 6a2a^ + aa^ + — .
■ 16
74
INVOLUTION AND EVOLUTION
IV. EVOLUTION — CUBE ROOT
Exercise 42
Extract the cube root of :
1. a3 + 9a2 + 27a + 27.
2. a«-15a^ + 75a^-125a»
3. a« + 6a^ + 15a^ + 20a3 + 15a2 + 6a + l.
4. 1 - 9 a + 33 a^ - 63 a^ 4- 66 a^ - 36 a^ + 8 a«.
5. 60 a^ 4- 1 + 240 a;^ + 64 cc« - 192 ar^ - 160 aj^ - 12 jb.
6. m® — 3m* + 5m^ — 3 m — 1.
7. 18 a* + 90 a^ + 125 - 3 a« - 31 a^ - 75 a + a\
8. m3 + m2+^+ ^
9. a« + ^V^^''^'
a^
3 27
4-
a^
10. a^-.^"'^ ■ ^^^' ^'
2c "*" 4c2
8c^
11. a^-3a:5 + 2a^-a:3^2^_«_l..
3 3 27
Extract the sixth root of :
12. 1 - 6 ?ri + 15 m2 - 20 m^ + 15 m< - 6 m« + m«.
13. a« + 60 a^62 ^240 a^b* + 64 6« - 12 a'b - 160 a^b^- 192 ab^
V. EVOLUTION— NUMERICAL
Exercise 43
Find the square root of :
1. 3969. 5. 15129.
2. 6561. 6. 93636.
3. 8464. 7. 1772.41.
4. 10404. 8i 2672.89.
9. .986049.
10. .01449616.
11. .01018081.
12. .000104101209.
INVOLUTION AND EVOLUTION
16
Find, to four decimal places, the square root of:
13. 8. 16. 2.5.
14. 14. 17. 37.561. 20. 1.0405.
15. 175. 18. .375, 21. .0035.
19. |.
Find the cube root of :
22. 42875. 25. 12977875. 28. .001481544.
23. 250047. 26. 28652616. 29. 34328.125.
24. 1860867. 27. 74.618461. 30. 20.570824.
Find, to two decimal places, the cube root of:
31. 9. 33. 7.3. 35. |.
32. 67. 34. 2J. 36. .0042.
Find, to two decimal places, the value of :
37. V5. 38. Vi5. 39. ViO.
41. V^.
42. v:oo7.
43. VIl2.
44. a/IO.
45. V.0017.
46. V2V2.
47. V5 -f V3.
48. V5-f-V3.
49. ^10-fV7.
50. Vl5-3^i8:
51.
40. V.9.
^ 5 + V5
52. J 10+4V3 .
^ V2
53. V.038 4-V.009.
54. Vv:5+\/A
55. vio+Vio+^10.
REVIEW
Exercise 44
1. Find the H. C. F. of 5 a* - 4 a^ - 64 and a' + a^ - 20.
2. Take ic^+3 from a^ — 2.x^-^Xj and multiply the re-
mainder by a; (a? + 3).
3. Find four terms of a^ -^ (a — 2).
4. Prove that (''+^)'-<"-^)' = a6.
4
5. Find the value of Va^ + 2/^ + 2;^ — (ic — 2/ — zy, when
a; = 3, 2/ = — 3, and z = 0.
6. What is the remainder if (a — 2) (a — 3) (a- — a + 5) is
divided by (a - l)(a + 2) ?
7. A certain divisor is a^ -{-x — 2 and the corresponding
quotient is ay^ — x— 1. Find the corresponding dividend.
8. What quotient will result if the sum of x^ — 5x^ -{-Sx
and 2qi^ — 5x — 1 is divided by the sum of 3 i»^ — 2 a; — 3 and
-2i»2-h4?
9. Solve ^^-^ + ^-=^=2.
x-\-2 X — 5
10. Prove that
("-fe?)^fe?)+(4ef)'=--
76
REVIEW 77
11. If m= r, ^ = n^ P —
a + 1' a + 2'^ a + 3
find the value of z hi h
1 — m 1 — n 1— i>
12. What value of x will make the expression
3(a; + 2)-4(a;-3)
equal to twice the value of a; ?
13. If -Ti = 9, find the value of — '
Sa—x x—a
14. Show that
5a;
-2a:-[4-i(a.- + 4)-21a;-3-(a; + 2)S-4]-^ = a
15. Show that
a(h-\-c — a) h(c-\-a — h) c(a-{-b — c) _^
(a _ 6) (c - a) "•" (6 - c) (a - 6) "^ (c - a) (b - c) ~ '
a-{-4:b 2 46 — g
.. .Q- r^ a-46 46 + a /2 IN
16. Simplify -^— X^--2^}
17 Ifg- 2findvalucof (^~^^^^~^^> (l+a)(l+2a)
17. It a- ^tmdvalueot jf^^^ (r=2^)
18. Solve ^-^^-1-^ + ^ + ^=0.
2g — a; a; + 2a
19. Simplify
6 («-!)(« -2)
a — 1 a + l
.O.Add4[^-l(ao-l)]a„<i3[^_l(c + l)-2].
78 REVIEW
21. Solve — ; ; — = m-\-n.
x-\-n x-\-m
22. Show that (a-\-b)(a+b-l)=a(a-'l)-\-2ab-\-h(b-l).
23. Find the value of — — a(x-^l) ^^ ^
IX X
when X
24. If a = ^^^ and c=^^, findcwhen 6 = -l.
62 a 4- 3
26. Simplify a-lb-o-\2a-2,-U3o-m,
a— J
when a = 1, 5 = 2, c = 3.
4a; .4
3i» + l T 3 5x + 2
26. Solve ic-
5 2 10 11 a; + 6
27. What must be the value of m in order that
6a^- a^- 11 a^-lOa-m
may be exactly divisible by 2 a^ — 3a — 1?
28. Show that ^(^-'^) _^JP±^^a when a = x-c.
29. Given that m = - r and ti = — - — -,
1 + a^ 1 + a^
prove that m^ + n^ = l.
30. Prove that — satisfies the equation
3
4a; — m x-^m __^
2x — m x^m~
REVIEW 79
31. Solve (a;-f3)(2/-2) = (a;-5)(2/ + 4) + 16,
{x — y){x + l) = l — x{y-x).
32. If x = ^^ and a = ^~^^ find x in terms of m.
33. Solve a(a — x) = b(x-^y — a),
a(j/ — b — x) = b(y — b).
34. When x = 2, find the value of
"-[-K''-^i)-|f-i)-<-8}-f]-
35. Solve (a-26)aj + a2 + 52 = (2a-&)a;-2a6.
36. What must be added to 2[l-3a;fl-2a;(l-5a;)j] to
produce -5-3x{l-2xy?
37. li x = — ^^ and m = ^ T" , find value of a; in terms of n.
w 2 '
38. What must be the value of m in order that x = 2 may-
be the solution of m{x — m)—S(x — S)?
39. Arrange (m — 2)* — (m — 1)^ + 3 (2 — m)^ in ascending
powers of m.
40. Solve ^-^= ^ + ^.
a; + 2?i a; — 2m
41. Expand [m- 3 n-f3(m-n) -n}]*.
42. Find the square root of
(m - nyi(m -ny-2 (m^ + n^)] + 2 (m^ + n^.
43. Solve ^-:=i^ + ^-ii^ + ^:::i-^ = l + ? + ?.
en cm mn m n c
80 REVIEW
44. What expression multiplied by itself will give tlie ex-
pression 4^rc6_i2a^ + 5x'-hl4.a^-llx'-4:X-{-4.?
45. Expand (a^ — 8cy and extract square root of the result.
46. Divide 1 ^^„ by 1 — —^ — — and extract square
(a-\-by -^ d'-ab + y'
root of the quotient.
47. Simplify V(a-2c)«.
48. Show that the difference between the squares of any
two consecutive numbers is 1 more than double the smaller
number. ^
49. Prove that q r is a perfect square.
a a^ — a- 4- a
50. Compute ^0.4 -f- V.004 to three decimal places.
51. Find the cube root of [2 a - (a + l-(a-2))J.
52. If a; = ^L+1, find the value of
a — 1
^(x + 4:){x-2)-2(x - 4).
53. Simplify \VlO — V5 to two decimal places.
-- oi a — 3a; b — 2x -,
54. Solve = 1.
64-a; a + x
55. Find the square root of
(a^_4a;4-3)(a^-9)(a^ + 2a;-3)
x^-^ex-{-9
56. Find, to two decimal places : ^ / "^ .
A/ V150
57. Solve 15y — 14:X = — 4:xy', SOy — IS x = 17 ocy.
EXPONENTS 81
EXPONENTS
I. TRANSFORMATIONS
Exercise 45
Express with fractional exponents :
1.
2.
Va.
<ra.
2V^.
5.
6.
7.
8.
9.'
10.
11.
12.
3 Va-".
V4al
13.
14.
15.
16.
^27 a^.
3.
^/a6Vd*.
4.
^32a:^2/'V^
Express with radical signs:
17.
aK
22.
abK
27.
ahl
32.
2 c^dl
18.
aK
23.
abc^.
28.
(ab)l
33.
a^^y.
19.
aK
24.
3 abhi
29.
1
34.
2ahhK
20.
«l
25.
ahl
30.
3 c^d^.
35.
4 c^d-.
21.
5aK
26.
ahK
31.
7cd\
36.
m »
5 a"6«.
In the following, transfer to denominators all factors having
negative exponents :
37.
ab-\
42.
3 a-^b-\
47.
4 a ^x.
38.
ah-'c-\
43.
a-'bc-\
48.
3-^m.
39.
2 abc-\
44.
a'b-'c-K
49.
9-kd.
40.
abh-\
45.
2-'a-\
50.
- 2-^a-^bc
41.
1 a~\
46.
3a'b-K
51.
-|a-'6c.
R. & 8. EX. IN ALG, -
-6
82
52
53. —
}
EXPONENTS
Write the
following without denominators :
a'
b'
54. 2^.
mn
^6 ^^'
KG 2 mri
2k,
c
66- !|-
cd
57. 2« .
59. ^«"
60.
4.C
61.
3a-^
a-'x-i ^ „^.
Express the following -vrith positive exponents :
62. 3a-\ _3_. 4VFJ
63. 2 am-*. ^ ^ «"'
64. a'^b-'c. 72 i^. 77 ^~'^~'
65. mn~^.
66. 2a-^6V^ 73. ^^=^- 78
67. om~^6"^c.
68. ab~^xy-\
tA,
(xZ-^
73.
2a6-^
Sm-^n
74.
1
75.
2-V
0-^6
2Va-2c-
79.
3 mn~-
4 a-icd-3
69. mn-V^^. ^_,^ ^^ ^-'ab'
70. a-^6-V\ ' a-^ SVa;"^
Find the numerical value of the following :
81. 4i. 85. 16^ 89. 125i 93. 4"^.
82. 9i 86. 27*. 90. (-27)i 94. 9~^.
83. 4l 87. 27*. 91. (-64)1 95. 16"^.
84. 9I 88. 8li 92. (-125)1 96. (-27)"i
97. 36"*. 98. (-32)-^. 99. -s/^^^.
EXPONENTS 83
100. ^J/(=:27p. 109. Sl-i.l;. 117. -(-2i^)-l
101. (^=:27/. jj^ 2-3-2-- "^- ^*)'*-
102. ^K ,,,; ,6-i".8i "'■ ^-«^"*-
103. (^16)'. ^^^ gi. j__ 120. ar^-^(#.
104. (<^^/. ' 32*' 121. 3-^. A.
105. 2-^.3->. 113- 9-* -81*. ,-j „j
122 T
106. 3-2.2-2. 114. 2-2.32.4.6-1. • 27^
107. 9-^.27*. 115. (2J)i 16-^ . 27"^
108. 16^.8"*. 116. (l^^^)"^. * 9-^.64-*
124. (4-3 . 3-" . 23) -^ (16-^ . 27"* • Sl"^).
Perform the indicated operations in the following :
125. w
« . a-\
130.
(i-« • a».
135. m-^-m^.
126. 0/
' . a-\
131.
a^.aK
136. 171^ -m^.
127. a
.a-8.
132.
a ' ai
137. a-*-i-a\
128. a'
^ . a-2.
133.
a-i . a-i
138. a-3-^a-2.
129. a-
-' . a-^
134.
aKa-i.
139. a-^^al
140.
a^cc . ax-\
147.
8*4-9i
141.
amhr^ . a^m"
-V.
148.
8-^-9-i
142.
2a62.3a-i6-
-1^
149.
a'h . a-162 . ah-\
143.
a« + 6«.
150.
2 a . 3 a^ . a-*.
144.
3a«-(3a)«.
151.
3 a^a^^/ • ohcx-\
145.
(a + &)«.
152.
o?^fx • a^ic-^
146.
(-2)-3-(-
-3)-^
153.
2Va-3aV^-a;"^.
84
EXPONENTS
157.
168,
163.
164.
165.
166.
167.
168.
x-Wx
154. a^Vx . a^Vx^ h- aV^.
155. 2a-^V^.3aiVx^^-ax-\
156. x'^aVc^-^x-^a^Vcd^.
159,
160.
VaV
a^\a^Vx
161
162.
3a;-^
a^a;
'VacVac
3 m ^Vo^
mn-
^Vm^
Sx'^V^ ■
a~^^ax~^
xr^-y/a-^x
2aWx^'
a'^y/d
c'Voc^
169.
170.
171.
172.
173.
174,
oT^x-'
2aV9a^
3 c-s/21 x--a-'
4
'V^TlV^
12/-
V4 a^x-
Keduce to the simplest form :
175. {aj. 178. (a3)-2.
176. (a2)3. 179. (a-^f.
177. (a2)-i. 180. (2a'y.
3x^-s/-Ua-^x
a/-27V64^
-^-125a-«'
181. (2a-3)-2.
182. (a-2)-^.
183. (4a-^i
(a-^-i
EX
198. (
PONENTS
212.
85
184.
(S/-32c^«)-3.
185.
(Sx^)-^
199. (
2-'a-^y.
213.
'\/-81-*a«.
186.
{x-'^)-\
200. {
;8-^)^.
214.
(9a-'y-r^.
187.
(5aby.
201. (
;-8-^)^
215.
(16a-V2/2)-l
188.
(a'by.
202. (
;- 27-^1
216.
{aV^'y.
189.
(a-'b-'y.
203. (
;-i25)-^.
217.
(a-'V^y.
190.
(ab-r^
204. (
;-8a2)i
218.
(a-WaV<)-^.
191.
(a*b'c-^^.
205. (
'2a-'c^y.
219.
(a-Va-^-l
192.
(646c-2)-l
206. (
[2ah-'y)-\
220.
(2a'^V^'y'.
193.
(-4a)l
(-aby.
(-2a'by.
207. (
208. (
209. <
[2m^np-Y\
221.
222.
223.
(ab-Wa-'by^-
194.
:^-sa-^y.
(aV4a-^)-3.
195.
[■y/16a'b')-^
(x-^yjxVwy.
196.
{-2aby,
210.
[2a-^Sa-y\
224.
l(Va-'by\\
197.
{-^a)-\
211.
[8a-2c-3)-l
225.
l(^Sa*by\\
226. ' l{</27'a-'yy\
227. yjie-'x^^.
228. (^[m^)^.
3/ ;
229. y-SVaF^.
230. (yjVu^y.
234. [(64ar^.^]-
235. ^j(16x'y-'Vxy)-^
236.
VV^Vm -\/-
mx ' m
231. (a-'yJ27arix^)-\
232. l^[(^</^Wr'f.
233. (^125 a^V^-^
237. (V27^)-2-- (V9¥-3)-i.
238. \m~^n^^mn~^yln^) '
^6 EXPONENTS
240. y25a-^bVx-''-r-y9a-'b-^Vs^,
3
242.
243.
244.
245. \xl^^^'\\^^^\~\
246. ^ M^;y6 c-3^g^y i^
Vc-' Vad-
1^-^
247. \\l- V". J27«V^|-'.
.48. [^^"^^-.g^^^J
249. (V •^^"'^'_ . ^332^i^F^r'.
I ^25a36-i^c2 J
252. iVa^ft-W^'c-iaVcVWaSc)
EXPONENTS 87
Collect;
253. (|)i+(^V)i + (32)i
254. (2)-2 4.8^-4-\
255. 3-2-27-^4-9"^.
256. (^?yVl6-f + A+(_2)-».
257. (3)-i_ 2-3 + (1^)^ + 128-1
258. 9-^ + 13 a;*' + 1"^ + {^)-\
259. ■^2TF'5_:^ + J-_128-*.
2"^ 8"^
260. 7aJ»-(7a;)«-17^ + A.
2
261. 8i-*-A- + :^-#
27-^ V34 9-^
Simplify :
262. 2". 22-^2". 271. (a^+i) V*)' ' («'"0~^-
263. 4 "-2 . 82-" . 2». m _n n
264. (4-. 2-) ^8. '^'- (-'^'^O-.
265. (af^-^-^i . (a^-r . ar=. 273. (3^+2 _^ 3 • S'*) ^ (9 • 3"+^.
266. {ahy+y -^ a'fb". 274. (a^^-^^ • a;-^«-*) -^ a;"'*-^.
267. a'"+" . a^"*-" • a""^.
275. S[(a)'»-i]-'-i ;""•+».
276. [K«^-r^r]-[K^m-
268. ar^^a^"*.
269. [(a'=+^)^-^ -5- (a^-*)^.
270. (x"'-^y(xP-''y(oif-^y. 277. (Va'^^-j-Va)'^^^^
EXPONENTS
278
279
2n+iy2 . 2"-3
1 "*^ 1
280. KC'""'0'"{ m2-l.^„»+l^
3n
a+6 g— 6 2a 1
283. ([ic *= ] . [ic~] -h a;~^)^
284
-. [^'{5)•(')(S)r■
II. MISCELLANEOUS APPLICATIOJSTS OF EXPONENTS
Exercise 46
Multiply :
1. a-2 - 2 a-^b-^ + &~' by a^ - b-\
2. a^ + aM4-2>^ by a^-{-b\
3. a-2-2a-^ + 3-a by 3a-^-2-2a.
4. a^ — aM + 6^ by a^ + ah^ + b\
5. a-3-2a-2 + 3a-i + l by a-2-3a-i-l.
6. 3a^-6a^ + 4 by a^ + 2aJ-3.
7. sJ-a^-{-2-4.ar^ hy '2a^-S + 2ari
8. ^-f.l^_-^ + 2-^by ^-2 + ^'.
V^ Va; Va; Va Va; Va
EXPONENTS %%
Multiply the following by inspection : '
9.
[a-' + iy.
18.
(a-2-3)(a-2 + 2).
10.
:a- + 3)l
19.
(a^+4)(a^-h5).
11.
[a-^-4.y.
20.
(5-«-2)(3 + a-^.
12. (
[a-' + h-y.
21.
(a^ - 6^)(a^ 4- 6^).
13.
[a^ + b-')\
22.
(a-^-3)(a-^-2).
14.
[a-^ + h){a-
^-h).
23.
(a-^_a-i)(a-^-2a-^).
15.
^a-^ + 3)(a-
^-3).
24.
(a-^6 + c-i)(2a-i6-c-')
16.
;a-2-4)(a-
-2-1).
25.
(a-i + 6-^ + 1)2.
17.
:a^-2)(ai
+ 2).
26.
(a-2 + 6-2_c)2.
Divide :
27. a-3 + 3a-2 + 3a-^ + l by a-^ + l.
28. a"^4-2a"^ + l by a"* 4- 1.
29. a + 6 by a^ H- &^.
30. a— 125 c"^ by a^ — 5c"i
31. a*-6a^ + 12a^-8 by a^-2,
32. x^-\-a~^ by a;^4-«~^.
33. a'— 3a^ + 3a"^-a"^ by a^ — cC^.
34. 10x-''-27a;-3 4-34a;-2_18a;-i-8 by ^x'"" -Qx'^ -2.
35 . 12 a"^ - 17 a"^ - 9 -f 13 a^ - 63 aHy 4 a"^ - 3 + 7 al
36.
6 a~i + 11 a ^ V » = -^-z + 10 ic^ by
Va Va
— + 5 a-^a;^ - 2 a;l
90 EXPONENTS
Divide the following by inspection :
37. (a-2 - 9) by (a-^ + 3). 42. (a - 8) by (a^ - 2).
38. (ci-2-6-2) by (a-i-6-i). 43. {a^ + 27) by {a^ + 3).
39. (a--* - 16) by (a-^ -f 4). 44. (a-^ - 64) by {a^ - 4).
40. (a - 81) by (a^ - 9). 45. (a"* - fe-*) by (a-^ - ft-^).
41. (a-'-b-^)hj (a-^-b-^). 46. (a--* - 16) by (a-^ - 2).
Factor :
47. a-2-6-2. 57. aj-* - 9 a;-^ + 8.
48. a-^-81. 58. x-^-Sx-'^-ASx'K
49. 4a-2_256-^ 69. 2 x'^ -{- x-^ - 10.
50. a^-9. 60. 4a;"^ + lla;'^-3.
51. a^-8. 61. a;^-27.
52. a-2 + 5a-^ + 6. 62. a;^ + 64.
53. a-2— 6 (1-^-^ + 5 2/-^ 63. a;^ - 8.
54. a^ — lOa^ + 25. 64. a; — 4.
55. a^ + 8a^.4-16. 65. Sa?-\-b\
56. m"^-5m"^-36. 66. a-i + 1256-'.
Simplify :
67. a-'^b-\ ^4 (a + l)(a-l)-i4-l
68. a-' + b-\ ' (a -\-l) (a -!)-'-!
69. a-'b + ab-\ 75. ^-(g + ^)"\
l + (c_l)-i
70. a-^bc + ab-^c + abc-\
71. a-\a + b) + (a^b)b-\ 76. ____^--^_^.
72. (a;-l+rO-^(a5-^-rO• ^^ a(a _ l)-i 4. 6(a + l)-i
73. (l+mn-i)-7-(l+m-^w). ' a(a+ 1)-^ + 6(a-l)-i*
EXPONENTS 91
78 mn-^ + nr^n ^^ x(l -[- a;)-^ + x-\l — x)
m~^ — m~^n~^ + n~^ ' x(l + a;)~^ — a;"^(l — a;)
80. r a-| + (m + n)-n |-j^ _^ ^^2 ^ ^2 _ a')2-'m-'n-'l
81. [(m + a)~\m — c)~^ + (m — a)-^(m + c)"^]
-s- [(m + a)"\m + c)~^ + (m — a)-\m — c)-^].
Expand :
82. (x-2x-y. 87. (V^-3V^)*.
83. (2a;-i + 3)^
84. (x-^-Sa^*.
88.
85. (a.-f2a.-0^ ^^ ^__^ ^^.
86. (Va;-A/a;)3.
Extract the square root of :
90. x'^ -10 x-^-\- 25.
91. a;-8a;* + 18ic^-8a;i + l.
92. 9a-2-6a-i + 13-4a + 4a2.
93. 9a;-*-30a;-3 + 67fl;-2-70aj-i4-49.
94. 4a;^ — 4a;^+13iB^-6a;^ + 9.
95. 9(B-12a;^ + 34a;~^-20a;~* + 25a;-l
96. 16a;-^--^7^-7 + 12^/aJ + 4^/a^.
■y/x
^^ 9 a 24Va 24 V6 , 9 6 , ^^
97. -7 ;= ;^H l-o4.
^ V6 Va «
92
I
EXPONENTS
Solve the following equations :
98. a;-i = 2.
105. a;~^ = -8.
112.
X~n = — 2.
99. ic^ = 3.
106. xi=^.
113.
--* = A.
100. x^ = -2.
107. a;"^=-i..
114.
»="* = t1t-
101. x-^ = 2.
108. x~^ = l.
115.
1
102. a;~^ = — 3.
109. a;"^ = 16.
116.
V5="^=iooo
103. a?* = 8.
110. a;" = 2.
117.
x^ = </K
104. a;^ = -27.
111. a;" = 2*".
118.
xt = V^.
119. (a; 4-1)' =
4.
127.
(a^
-l)-2
= i-
120. (a; + 2)3 =
125. 128.
(2
0.-1)-
•'=.v
121. (a;-l)^=
= 3.
129.
(X-
-^ + 1)-
'' = 9.
122. (a? -5)^ =
= 1.
130.
(X
■^ + 2)-
■^=16.
123. (a; + l)^ =
:4.
131.
(a.-
■t-5)-
-^ = i.
124. (a;-3)i=
z8.
132.
(a:-
■^-i)-
■'=h
125. (a; + 4)-3:
= 27. 133.
(X-
-^ + 3)-
-" = 1.
126. (3aj-l)-
2_
:i 134.
(X
■|_7y
= 1.
Find the value of
X in the following :
135. 4* = 8.
138. 9^ = 27.
141.
8' = ^T.
136. 4* = 64.
139. 9^ = ^V
142.
32- = 9.
137. 16^ = 8.
140. 27* = 3.
143.
a)-^=8.
Find the value of x in the following :
144. x-^ = y', y^ = 4:. 148.
145. x-^=y', 2/^ = 2. 149.
-^ = -8. 150.
-"> 151.
146. x'^ = y-^',
147.
3.-1 =2/* J 2/~^ = -2.
«"^ = y-^ ; 2/f = 4.
a;"^ = y~^ ; t/ = '^'
BABICALS 93
Find the value of n in the following ;
152. 2«-i = 16. 157. 3"+! = ^.
153. 3'*-i = 27. 158. 4'*-2 = J[^.
154. 9"-^ = 27. 159. (i)'*-' = ^.
155. 4"+' = 16. 160. (i)"-' = A-
156. 16~-^ = 8. 161. (1)"-' = ]^.
RADICALS
I. TRANSFORMATIONS
lizercise 47
Reduce to the simplest form :
1. V8. 14. --v/128.
2. Vl2. 15. -v^^Si.
3. V20. 16. \/32.
4. V28. 17. -^162.
5. V27. 18. --^96.
6. V45.
7. V48. 20. Vo^.
8. V72. 21. VaFb\
9. -Vl25. 22. Va^¥?.
10. ^16. 23. -Vl8^.
11. ^24. 24. V27V-
12. -a/54. 25. ^54^i¥.
27.
28.
-■v/320a%V.
3 V27 a.
29.
4V28a«6^
30.
2 V56 m%3^
31.
V20 m^*.
32.
-2-v/250a*
33.
^16 a^^«2/^«.
34.
35.
36.
3-V/64 m^n.
i V54 a^
37.
-^V20 6c^c?.
38.
-v/54 ai«6^.
13. V108. 26. -V128mV. 39. -\Vi25<^',
94
RADICALS
40. i/27a''3^.
41. aV(a + c)^
42. -aV3a\a-{-iy.
43. 5mV(a-l)l
44. -a^a%a-iy.
45. Va« + 2a2a;_{_aa;2^
46. V36(a2-»^)(a + x).
128a^
9c2
48
■V
49. m^
50.
aV
25 ar^
3^
108^
49 c^
63
64.
65.
Change to entire surds :
51. 2V5. 54. 2^/5.
52. 3V7.
53. 4V3.
2 3/9^2
a'
a /-
3 /o-
— V2a.
m
Keduce :
73. Vi
74. V|.
75. Vi.
76. VS.
77. V|.
78. V|.
55. 3^4.
56. 2^7.
66.
67.
3a
2
57. 2aVa.
58. 8a-Va.
60. 3a2^a2
61. 2a-\/3aP.
59. -2a Va. 62. -2aV5a.
"2^.
69. -^'^9.
2_a3/_3_
3\2a
68. -^V^l
70.
71
72. -(«-l)^5?I•
79. V|.
80. Vf.
81. Vf.
82.4
'a?
83. Vf-
84. ^.
85. -^J.
86
87. -
a4-2>
a + & 3
3/ n~
88.
89.
90.
2Ai^.
BADICALS
95
91. -3^^.
4: ax
27 *
92. i^
93.
95.
96.
_9_ /4_ac
2a^' 3 '
99
A/l2a2
« 3 aV
3\^-
94. -AJ/-^
3a ^ 8
Simplify the indices of :
2c^ 8 1 27 a*
3a\ c2 *
97. iV^.
100. V6|(m-n)^
101. -y/^ia+iy.
4/ 9"
103
104. -Vc^.
105. a/9.
106. -v^.
107. -^^25"^^.
108. ^?d^V^.
102. VW^^^^
109. -v^QoV^.
110. -^/Sl aVd".
112. V27aV2.
Change to radicals having
114. V3 and a/5.
115. V5 and -y/W.
116. a/9 and -s/l,
117. Vl5 and -v/SO.
122. Vm,
Which is the greater :
123. 2V3 or 3V2?
124. Vil or -\/30?
125. 2V3 or a/42?
Which is the greatest :
129. V5, a/10, or a/IS?
111. Vl6a*b'c\
113. a/100.
the same index :
118. Vn and A^SO.
119. a/25 and a/75.
120. a/6, a/15, and a/35.
121. A^, a/7, and a/10,
a/w?, and a/wi^.
126. 3a/5 or SVS?
127. 2a/4 or a/10?
128. a/| or a/|?
130. V6, \/16, or a/35?
96
RADICALS
Collect :
131. V50H-V18-V8-V32.
132. Vl8~V98+V50-V72.
133. V27-VI2+V75+V3.
134. Vl2a-V27a-V48a + Vl08a'.
135. Vow^ — Va%i + VOom^ + V4 a^m.
136. |Vl2-V50 + ^V48-Vi8.
137. V20-V| + V| + 4V2-3V5.
138. Vl24-Vi-V27 4-V|-Vl08.
139. V50-^-6Vi + 3V|.
140. -^i-Vi + V98-2V27.
141. 2-v/| 4- 3-5^ -2^/144.
142. 2V|-3VS-V}4-Vi000.
143. 3Vo^ + 4
25
4/aV
81
144. V50-v''432+V32 + ^250.
145. 3^4-^24-^3+A/i6.
146. 30VJ-fV8 + 9V84l.
147. VT80-2V5 + 15V|.
148. |Vl62 + 10V4|-13V2.
149. -t/36-V|-4V6 + 2VJ.
150. 10Vl2} + 7V2-3V338 4-5^ + 4Vf.
151. V24-6Vi + iV96-V66| + |V¥-
RADICALS 97
152. 6V33j-V96 + V|-|V-^ + 4V6.
153. 12Vi6j + 5V3-5V432 + 6V^.
Multiply :
154. (V20-fV80+V45) by V5.
155. (V8-2VI2 + V2O) by V6.
156. (^-1^32 + -J/5) by ^/16.
157. (2V3-2)(2V3 + 2).
158. (2V5 + 3V2)(3V5-4V2).
159. (5V3-2V2)(3V3 + 4V2).
160. (V2+V3)2. 163. (3V2-5V3)2.
161. (V3-2V2)2. 164. (V3-V2)^
162. (2V3-2V2)2. 165. (2V2-2)^
166. (3V2-2V3)3.
167. (V7-V2 4-V5)(V7+V2-V5).
168. (3V| + 3Vi-10Vi)(iV24 + iV75 + V20).
169. ( VlO 4- Vl9) ( VlO - Vl9).
170. (Vl3 - 2 V22) (Vl3 4- 2V22).
171. ( V2 -\-x-{-Vx) (V2Tx).
172. ( Va + 1 - 2)2.
173. (2 Va^"^^ - 3)2.
174. ( Vm + 1 - Vm - 1) ( Vm^^).
175. ( Vm + 2 -f- Vm) ( VmT2 - 2 Vm).
R. & S. EX. IN ALG. — 7
98 RADICALS
176. (m2 + mV3 + 3)(mV3-3).
177. (Vm — Vm — n + Vn) (Vm + Vm — n + Vw).
178. (Va-l+Va + 1)-.
179. ( V2I - 6 V3) ( V2I + 6 V3).
180. (5V?T^-4V?^^2.
181. (2V^ + V4^^)(2v^-V4"=^).
182. (Vo+I - V2)(2VaTl + V2)(2a+ V2a + 2).
183. VS--\^-W- 185. Vi|--v/W-
184. VS--\^||- 186. (V2-2^/4)(2V2-^/4).
Divide :
187. 2V32 by 3Vi20. 193. \V^ by ^V^.
188. \/8l by V3. 194. (5V18-3V27) by 3V5.
189. ^J/12 by 4V2. 195. (2V54+iV24) by 3Vi.
190. VU by ^32. 196. Kk by Ajji.
5 Vc "^ 10\a
191. -Wc^ by -ySom^. r-
192. (^12 + 4VI8) by 6V2. '''• (IV^-^V^ ^^ ^•
198. (12V5 - 8V15 + 3 V30) by 6VIO.
199. 10V3 - 15V42 - 9 V2 by 5 V6.
200. (5^^^ 4- 3^/45 + 6^/30) by 2\/i8.
201. (</32--v^'48-a/80) by </3
202. Xi^, by f-^^j:K:Y
RADICALS 99
n. MISCELLANEOUS APPLICATIONS OF RADICALS
Exercise 48
Extract the square root of :
1. 6 + 4V2. 7. 44-16V7. 13. 16 + 2V39.
2. 11 + 6V2. 8. 30-12 V6. 14. 74-6V77.
3. 28-10V3. 9. 88-16VI0. 15. 77-28V7.
4. 21 + 8 Vs. 10. 57-f-12Vl5. 16. o?+h + 2a^h.
5. 45-20V5. 11. 207-40Vli. 17. o? + 2c^-2a^/2~c.
6. 42 + 12 V6. 12. 82 + 12 V42. 18. m"^ -\- m + 2 m^Vm.
19. 2a; + 2V^^^. 20. m^H- (2m-2)V2m-l.
Rationalize the denominators of :
,1. A.. 29. -1-. 37. 2Va + 3Vc-
V2 V3-2 2Va-3Vc
22. A. 30. -^ 38. 5V2^-V6j.
V3 V2+V3 5V2a; + V6a;
23. A,. 31. 2 + V3 g^ 4V2^-3V^ ,
V2 * 2-V3 * 3V2^ + 5V»
24. 11. 32. ?V|±2 ^^ aVc-ftV^ ,
' V6 * 2V2-1 * aVc + 6Va;
V24 ^, V5-V3 a + V^^^=^
25. -^^-^^- 33.
41.
V3 V54-V3 * a-^d'-l
26. ?^. 34. ^^ + ^^ 42. V^-3+V^ .
Vi2 * Va-V6 * VaJ-3-V^
27. -^. 36. 2V3 + 3V2. ^3
2+V2 2V3-3V2
28. ^. 36. 3^-2^^. 44.
V3^
; + l+V2a-
-1
V3^
' + 1-
-V2a-
-1
Vm^
-2-
-Vm2 + 2
3-V3 2V5-2Ve Vm2-2 + Vm=^-f-2
100 RADICALS
Find, to three decimals, the numerical value of:
45. ~ 47. -^. 49.
V3 2V3 2^2
46. A. 48. A. 50. ^^^±1.
V5 -v/2 V2-1
53. ?V|±3^|. ^^^
51.
V3 + 1
V3-1
^9.
V3 4-V2
V3-V2
5V3 4-
■2a/2
2v3-2V2 V2-V3
Simplify :
55. Vl5-6V6.
V8
(V6-V2)(V3 + 1)
58 Va; + 2 + Va^-2
59.
Va; + 2-V«-2
1 . 1
a — Va^ — 4 a 4- V a^ — 4
60. ^ _ + 1
(3-V2)2 (3+V2y
g^ 2V^-i ^ 3V^"=i:
3Va; + l-2Va;-l ^^x + l+2Vx^^
^^ V26 4-8V3 _
V6- V2
63. :^^19-8V3,
V3 + 4
KADICALS
101
64.
65.
66.
67.
68
Va 4- V^ Va — Vb
2-Vb
2Va
V3-I-V2-V2
V3 + V2 + V2
a; + 5 ic — 5
( Vll + 6V2) + (^6 + 4 V2) ^
5 - 2 V2
\ ^ a* a ^ a; ay
70. [3V3+(V28-16V3)]2.
71.
72.
73.
74.
(a; + V4-^(a;-V4-arO
V34-2V2-V3-2V2
V3 + 2V2+V3-2V2
3-V2
Vo; 4- 1 + Va.' — 1
Va; + 1 1
Va;-1
34-V2
V3+V3+V2 V3-V3+V2
a + Va r a-4 (a4-2)Va "|
« (« + 2) LVa - 2 a H- Va J
[Hint: a - 4 =( Va + 2)(V^ - 2).]
75.
102 RALICALS
76. Show that — - — = 2 + V2.
2-V2
77. Show that ^ + ^^~^^ =V5-2V6.
2_V2 + V6
78. Show that ( V24 + 16 V2 - Vll + 6^/2)^ = 5.82842 +.
79. If a; = 4 + 3 V5 and 2/ = 5 + 2 V5, find value of (a^ _ /)2.
Solve:
80. V^+T = Vl2. 85. 2(V^-3)(V^ + 3)=3.
81. V«T3=V2a;-10. 86. Vx^^ - V^^^ = V2.
82. 3Va;-9 = 2Va; + ll. 87. Va; + 3 + Vx- 2 = 5.
83. 2V^T3 = 3VlO-a;. 88. V3H-V9i» + 1 = 3V^.
84. VflJ 4- 13 = 13 — V^. 89. Va; + 9 mw = 3 m + V^.
90. </10a;-6=V2.
91. Va; + 3+V9aj-l=V4a;-l.
92. V2a; + l-V8a;-l=V2a;-l.
93. ^j7 + ^^S + ■^^ = 3. 95. ViC + 7+V^ = ^.
94 V^ + 3 _ V^-f-9 __^__
V^-2 V^ + l" 96. V2+V^^^ = V^^^.
97. — Va; — 3 = V^.
Va;-3
98. V^ + V^"^ -5__ = 0.
99. 2a^ = ^ 4-V4a;-3.
V4a;-3
IM AGIN ABIES 103
100. V^^^ + V^ = _6.
Vx — 5 — Vic •
VaJ + m — V^ _ 1
Va; + m + Va; ^*
102.
103. V2a;-a2-f-V2a; + a2 = 26.
104. Va;-2a= ^^ - V^.
Va; — 2 a
105. V2^ — Vm = V2a; — Vm(5m + 8a;).
106. J^- J^ = 0.
107.
a; + 4 ^a; — 4
3VS + 2 3V^+1
2V^'-7 2V^-5
108. 2-\/x - 2 - 3 Va; + 2 = 5 V^.
109
Va , « /-
Simplify :
IMAGINARIES
Exercise 49
1. V^=^. 5. 2V-81. 9. -2V-36a«.
2. V^16. 6. V=^. ^^- V-(aj + 2/)l
4. V^lii. 8. -V^9^^. ''• -^^-(" + ")^-
104
IM AGIN ARIES
Collect:
13. V
14. V
4 + V-94-V'
"9 4-V^^^36
25.
V-49+a/^64.
15. V^^36 + V^=32r-V^^100-V^169.
16. 2V-9
17. 3V^=^-V
3 V^^16 + V-49 - V- 25.
V^^144+V^256.
81
18. iV
4 H- 2 V- 36 - \-\/- 25 + iV^^ST
19. 2V^ + 3V^
v
2V-
^ + 3V-^V
20
+ 2V-4a2-V-9a2 + V-16a2.
21. V-4m2-iV-16m2-h2V_i21m'^-V^
m''
22. aV
Multiply :
2
a
16 a^
23. V^^ by V^T.
24. V^=^ by V^^.
25. V^^ by V3.
26. 2V^^ by V^^.
27. 3V^^ by V^^.
28. V^^ by 2^^^.
29. 3V^^ by 3V2.
30. _ 2 V^=^ by V5.
31. 2V^^ by -V6.
32. 3V^=^ by 2V^^.
33. 3V-27 by -V^^.
34. - 2 V5 by 3 V^=^.
35. a^—a by Va.
36. — aV— a by — V— a.
37. -2V-3a by 3V^^.
38. — 3aV— a by 5V— a^
39
40
— 2aV— 2(1 by aV277.
-3aV"=^by -2aV^=^.
41. V^^ by V^6 by 2V^=^.
42. V^=^ - V^^n^ + 2 V^^15 by 2 V^^.
43. (3+V^ by (3-V^^).
IMAGINAEIE8
105
44. (o-V^(5+V^.
45. (2-V3)(2-V^.
46. (2+V^(2 + 2V^^).
47. (5-2V^^)(3-3V^=n[).
48. (3V2-V"^^)(2V2-3V^2).
49. (4-3V^=^)(2+V^^).
50. (Vir3_v^(V^^+V^^).
51. (2V^5-3V^(3V^=^ + 2V=^).
52. (3V^=^ + 2V2)(4V^:^-3V2).
53. (3-V^)^ 54. (2-3 V^'. 55. (2V3-2V^'
56. (a-l-V^l)(a-l+V^.
57. (a + &\/^l)(a-6\A=3).
58. (2 a + 2 6 V^^) (2 a - 2 h^^l).
59. (V^l+V^2-|-V^^)(V^=i:4-V^=^-V=^).
60. (V^^-V^=^ + V=T)(V^=^+V^^-V^.
61.
62.
63.
Rationalize the denominators of
3
12
V^=:3'
64.
65.
66.
V-15
2V3
3V^=^
2V^
67.
3V-9
-6V-3
68.
Vio
V-2
RClk
Vl5
2V-3
70.
71.
72.
-5V12
Va
73.
1-V=2
74.
34-V-2
75.
V2-Vi:3,
V2+V"^
106
IMAGINABIES
76.
77.
78.
V3 4-V^2
1-V^ *
vCis + v^
79.
80.
g + V-l .
a-V^l*
m + nV— 1
82.
83.
m — n-
2a-|-6V^
3a-26V^^
2a-6V^ri
2-3V2 e, V2a+2V-2a ^^ a-vT=^
ol. ^ — — — — • o4i
2V-2H-3V2
Va— V— a
a — Va — 1
Simplify :
85. (V^l)*. 87. (V^*. 89. (V^)-*.
86. (-V"=3)^ 88. (-V^=^/. 90. (-V^^)-^
91. (1_V^'-(1-V^^)^
92. (3-V^'-4(5-V^^)'.
93. (l+V^'-Cl-V^'^.
94. (1-V^^
95. (1_V^^-(1-V^^+(1-V=^).
96. 2(V^n:)3-2V^(V^=3-l)^
97. = (l--\/^^y. 99
V2-1 ^
98. (1 + V^)--(1-V^1)-. 100.
2V^1
iC+V — 1 iC— V— 1
m + ^V— 1 , m — nV— 1
Find the square root of :
101. 2-4V^^.
102. 1-56V^^.
Kesolve into imaginary factors :
105. a + b. 107. a + 2 6.
106. a + 4. 108. a2 + 4.
m -- 71 V— 1 wi + n V— T
103. 32-32V^.
104. -3-12V^^.
109. a^ + 1.
110. 2«2 4.3.
GENERAL REVIEW
Iizerciae 50
1.
3.
Find H. C. F. and L. C. M. of :
J m^ + 6 m" 4- 5 m — 12,
[m^-3m^-22m-12. 5.
6m3-llm2-14mH-24,
8m» + 18m2-llm-30.
lOa^ + a^-hlSa'-S,
16a^-a'-\-2.
l-8a;-3,
l + 9x-^-22ar*,
4.
rioa^
l8a^H-4a*^-2a2»_l.
1 3 a-1 - 4a~* - 13 + 14a^,
Simplify :
2 mn(m -^ n)~^ — m . 2 mn(m -\- n)~" — n
n-^ -\- (m — 2 ny^ m^^ -{- (n — 2 m)-^
i-1
9.
(g - 1) [3 g + (<^ - l)n-^-(l-3 a-{.a^(a^^l)-^-(a-l)-\
(I_2g + g2-2a«)(l + 2g + :^g2 + a3)-i
10. [(m — n)(m + w)-^ — (m + 7i)(m — n)-^]
11. 1+ K,
-W-2).
2-
1 +
1
2 +
3-2a?
a;-l
12. If - = ±, show that
n y n y
107
1^ 'T-
GENEBAL REVIEW
1 1
^-^^ + 1^-^
14. (a-l)(a + 3)-i-l-(a-2)(a-3)-i
+ [5(3-2a)-a2][9^a2]-^
15. m^-f
7M'
m2 +
^/i""
??i^ + m^
mr
16. [a-6(c-a-i)-i]-\
17. If « = ^, show that '^^ + ^ = _3m±^.
6 n 6a-f-36 5m + 3n
18. Show that
a^ — 2/^ ' (c — 1
Solve :
1 +
19.
3ic4-l
x-\-^
x-1
ir + 4
1 +
xy
ix-yf
2a^ + a;-3 x-2x' + Q 2-^x^o?
20 3a;-l a; + l , ^a; + 4 ^ 4a; + 5
8 9a;-16 4 6
21. ^±l-f.(a;-l)(a;-2)-i ^^~^
cc + 2
x-1
6
23. 5a;
= .-[3-{
aJ +
a;-3
15 *
2
(7 a; -12)-^ [_' [ ' (3-a;)-M_
24. [l+a;(l-a;)-i][l-a;(l4-a;)-'][l-a;'+(l-a;'K^]=3.
25. Show that -^ satisfies «-4a; _ a + 2x ^^
5 2a — a; x + 2a
GENERAL REVIEW
109
26. Solve (m + x){n-\-x)~m{n-{-p) = '^^ + oi:^.
X ex
27. Show that cd + -, = — + « when x = ad.
d a
Solve the following
28.
29.
30. <
31.
= 3,
32.
x-5 2x-y-l _2y-2
4 3 5 '
2y-^x — l _x-\-y
9 4 *
x — 2 x-\- y—1 _
3 4
X -\-S x — 2y—l _^
4 2
2a; + l 2-Sy ^l
3 5 6'
3y-2 2a; + 3 _3
4 6 8*
( 4:X — Sy±l __x-\-Sy
I 2^ T"'
3a; + 2y j^^ 2a;-3y
5 3
I ^x-V±y _ 2 x + 1.
3 ^ 2 '
14 ^ ^^ 2
36- -I
37.
38.
39.
f^ i_^_13
2a;"^32^ 3*
13a; 2y 27*
?/ a; 3a
1+1=A.
x y 2a
Sx-{-4:y — 5z = 2,
2x-Sy-Sz = -9,
Sx-y-2z = 0.
(2x-y-\-Sz = -i,
40. -! Sx-3y-2z = ^,
\4.x + 2y-5z = 5i.
( cy + bz = 2,
^^ (ax-by = a^-b^-2ab, ^^ I ^,_^,^^2,
33. 2aj + 32/ = 3a; — 22/ = l.
35. i
bx -\- ay = 2 ab -\- a^ — W.
a-^b a — b
4a6
bx + ay = 2.
[ x-'-y-'-z-' = S,
I y-^-x-
110 GENERAL REVIEW
Expand :
43. {2x-ay. 44. [3a - {2a+ (5a-2^"=l)i]^
45. ( "^-^^ Y. 47 ^..--^^Y
46. {a-^ + 2x-y. 48. ("v^ ^-Y-
Find the square root of:
49. a;-'^ + 4a;-3'»-2.'B-2'»-12a;-~-f-9.
50. a^-2a2»'-ll + 12a-2- + 36a-^.
51. 4-4a^-lla + 14a^ + 5a2_i2a^^.4aS.
52. (a;-2 - 4) {x-^ - 3 a;-* + 2) (a;-^ + a;"^ - 2).
53. (m + m-y-4(m-m-i).
Find the numerical value of:
54. V.073 to 5 decimal places.
55. V.0073 to 5 decimal places.
56. Vs - 3 VXiM to 3 decimal pla<;es.
57. V.007 + .3 VlAi to 3 decimal places.
Simplify :
58. ^!i^\^/:r2iw.
59. [-3-v/-27a-2-^(- 8-')-i]-'.
64. f-s/-S(^d-
65. 64-^-flV'</Sr^^
GENERAL REVIEW 111
67. (SaV^-h27 x^Vcir^yi
\ 16 ac'VdJ
68. (a?"*)"-^ . (a^)'»+l(a;'")l-2^
r
n+l
_ _ 69. Jinh^.x^Va'n
66. (m^) « (m^) ^ -- (m-y\ \^ '^^ \ A^-v/n'
70. V25 a-ift V^(27-ia"V^^aa;^6-i)i
71. [7a;-V^]« • [3a^2/"*2;]-^ "^(s^^^T*
Collect:
"■ (1)^3^
73. 7-1 + C^y - (7 a;)« - 49"^ - 2-2.
74. 81-^-5a;« + 9(3)-3+(125-V^ + rt
75. (^:^J_V3T27^ + (-243)i + (:^J-
Multiply :
76. i-l + ft'^by a-2 + l+r^-
77. a3-3 + 3a-3-a-« by a'---^---
a a*
78. J^_4aj-i + ^-24by ^+4^ + 5.
Divide :
79. a-^—b by a"*— &i
80. 27 m-3 - 8 w2 by 3 m-i - 2 nf
81. a;^+2a;^-16a;"^--32a;-^ by a;*+4a;'^+4a;"^.
82. a--^ + a-2'"62n _|_ 54n by 1 _^_j_?,2n
112 GENERAL REVIEW
Simplify: 83. (oT + 2 a'^^y - {a'^ — 2 a-'^f.
84. [(a + 6)^ + (a-6)^]2.
85. [(7/i-l)*+(m + l)^J.
3x _35
86. (m'^-m ^)-^(m' -^l-[.m-'').
Collect: 87. f VlGj + ^ - yV V432 + ^ VIp.
88. 5V75--jVi47--^ + 2VJ-^81.
89. Multiply ^ by -^^ by V|.
Simplify: 90. (3V6-2V3 + 5V2)2.
91. (2V5-3V3-2V2)(2V3+V5-3V2).
92. (^9-6V| + V48)2.
94. (V3-V2)2-2(2-V6)(2-V2).
95. (3V2-V3)(2V2 + V3)(3V3-V2).
96. (?^-5^ + iO.V^25-^216).
VVlO V6 V2/ ^
97. V52-6V35.
98. V2a + l-2Va2 + a.
99. (^/2-V5)^. 102. (V2 + ^3--^)2.
100. (V2-^y. 3^-4^81
101. (2V2- 3^2)2. ^^^- -^243
GENERAL REVIEW 113
Kationalize the denominators of the following :
104. -^ -• 106.
3V6-2V3 ' V3-V2 + 1
105. — ^- -. • 107. '^ + "^ -
Find the numerical value of:
^ V5 + V2 V2
110. Which is the greater, V5 or -^/\l 9
111. Which is the greatest, V|> \/f, or \/|?
112. Show that V| > ^f .
Simplify :
115. If m = i ( Vc + d + Vc - 2 d) and
yi = ^ ( Vc+c? - Vc-2d), find the value of m^ + n\
116. Simplify by inspection :
(Vm 4- 71 H- Vm — Vn) (Vm + ti — Vm — V^).
117. Change^— J^i-II-^ to an entire surd.
c — cZ ^ c -f fZ
118. If a = 11, 6 = — 12, and c = 3, what is the numerical
value of , /-To A —
— 0— Vo^ — 4ac ^
2a
119. If 71 = 11, a"=5V2-2V3, d = -(V2+V3), find
value of ^[2a + (7i-l)d].
R. & S. EX. IN ALG. — 8
114 GENERAL REVIEW
Simplify :
■« X — 1
x + 1
Vx — Vy -y^ + V^
122.
123, f^^^ — h — Va + & Va — 6 -f Vg -|- 5
a + 6/
\ Va — 6 + Va + 6 Va — b — V
Solve the following equations :
124. V» + V^^^ = V5. '^' »'
125. V3a;-2 4-V3^-2 = 0. <) ^ '^t^
126. Vic + 6 + Va;-4 = 2. ~ U » £>
127. Va; + 2 + V4« + l = V9a; + 7. 't -^ "^
128. VS^+T - V2x + 3 - V2a;-2 = 0.
129. V2 + V4^+5 = V2¥T3.
130. -^+ 1 2
V^^=^ Vx+^ Va^-4
4
131. -Va;-V4+^ = 0.
V4 + a;
132. ^ 4- ■'^
aj + Va^-3 a;-V^233 3
133.
134.
GENERAL REVIEW
2-x
X
V2 + Va; V2-Vx
2 m — n V2m^ + n
V2
mx — n
x + n
Simplify
135. (V^ri)3+(V^/.
136. (V^=^)'-(V^^
137. (l_V^2-h(l+V^^)2.
138. (i-2V^^y + 2(2-^^iy.
139. (V3 + V^)(V3-2V^).
140..^(2y3-3V^^
141. (-|+iV33)3.
142. V4 V6 - 11.
Rationalize the denominators of :
V^ _. 2V18
115
143.
144.
V^18
V2
145.
146.
-V28
147.
148.
V5-
32+2V^
What are the conjugate imaginary factors of :
149. m-{-2n? 150. a^ + T? 151. 3^2 + 2?
116 QUADRATIC EQUATIONS
QUADRATIC EQUATIONS
I. NtBIERICAL QUADRATICS
Exercise 51
Solve :
1. 2x^-7x = W, 11. x'-Q^O.
2. 2x'2 + a; = 15. 12. x'-4.x = 0.
3. 3a:2_^7^.^20. 13. a^ + l = 0.
4. 6x^-19x = S6. 14. 3a^ = 7.
5. 5x2 + 14aj = 3. 15. 5ar'=llx.
6. a;2 + 3a;4-l = 0. 16. 2a;- + 3 = 0.
7. .T2 + 3a;4-3 = 0. 17. 5ar-3x + l = 0.
8. aj2_5a;_i = o. 18. 3«2_^5^,_^3 ^q
9. .T2-5ic + 7=0. 19. 5a^ = 2a; + l.
10. 3a;2 4-2iK + l = 0. 20. 7ar = 6x-l.
21. 3ar^ + .^'-5 = a;2-ll + 8a;.
22. (2x-^S)(x-5) = (x-5)(x + S).
23. (3aj-7)(2a; + l) = (5a; + 2)(2a;-3).
24. (2a; - 1) (3a; + 5) - (a; + 5) (3ic - 2) = 5 - (« - 2)1
25. (3x+ 1) (a; - 5) - (2a; - 1) (3a; + 2) = (a; + 6)^ - 1.
26. (x -5y-(2x- 3)2 _ (a; + 4)2 = a; (a; - 5).
27. (2 a; + l) (a;- 5) +2(a;- 3)2- a;(a; -4) = 2(a;-|)2-15|-.
28. (a; - 5)2 - (3 - 2x)2 - (2a; - 1) (a; + 4) + 5a; = 0.
29. 2(a; + 2) (3a; - 1) - 3 (a; + 1) (4 - a;) = x(a;- 2) - 17.
QUADRATIC EQUATIONS 117
30, ^ + -^ = 1. 40.^-1 = ^.
x — 1 x-^1 x + 2 a; + 4
31 ^-1 1^ 1 41 2a;-3 a;-! ^ .^
*a;4-la; 6* *3a; — 2 x
32. ^-4 = _^. 42. 1-1=1 - + 2
3-a; 5 9-2ic 3 x 2a;-l
33. ^^±^-^:z2^ii 43. _J ^ + _5_^0.
a;_2 a;4-2 ^ a;-l cc + l 2-a;
34. 2^^ + 2^+5 = 2. 44. -5 ^+-A_=0.
2a; + l a; + 8 2a;-7 a;+4 a;+7
35. 4^Zli + §-l = 0. 45. _2_= J0_ _^.
2a; + l 2 a; x-2 x-\-2 x'-4.
3g _^ fl^-1 ^2 46 ^-3 24-a; ^ (a?+iy+4
x — S x-\-3 ' ' 1—x 1+x 1—x^
2^ a; + 3 3x-2 ^^ ^^ 2x-l x-2 ^ x-S
*« + 5 aj — 5 ' x — 2 aj — 3 a; — 4
^„ 2a;-l a;-2 q ^o 2«-1 a;-7 . 3a;-l
38. T = o. 48. -=4 — .
X x-\-l a^H-l a;— 1 x-\-2
39. ^Zll + ^Z:5 = 3. 49. ^±l-^^±?+^!=5 = 0.
a;-2 a;-4 a;-3 .t+3 ^9-a^
50 2a;4-l -^ x — 4. __ —7x
2x-8 2a; + 3 9-4ar'
2 a; — 3 a.' + 13 _ 13
a; — 5 a; — 3 ~~2a; + 5
52 5 a;-1^ 2(a;4-3) ^
2a;-l a; + l 2a; + 3
53 a; + l ^ 2a;-3 ^ 36
3a; + 2 3a;-2 4-9a^
QTIADBATIC EQUATIONS
60. 2V^=Va;-3-f 3.
61. V2x-l = Vx-\-l.
118
54. 3VaJ + 2 = 2a;-5.
55. a; + 5 = 2V5 oj + l.
56. V3 a; + 7 — a; = 3.
57. 2V3 a;-f-4 = VSa^-S a;-4. 63. 2Va; + l - V2a; -f 3 = 1.
58. 2V^ = a;-3. 64. 2 V3 a; + 7 + 1 = 3 Va? + 3.
65. 5Va;-l-3V3a;-2=-l.
62. Va;4-l-l=V2a;-5.
59. 2 Va? — 1 = a; — 4.
66. 2V3a;-2-3V^+3 + l = 0.
67. Va; + l+V3a;-f4 = V5a; + 6.
68. Va; + 5-V2a;-7= V5.
69. 2 Va; + 1 + Va; - 2 = V7 a; + 4.
70. V4a;-3-V2a; + 2=Va;-6.
71. ■y/Sx-5-{-Vx^^ = 2Vx'^^.
72. V2-3iB-V7 + a;=V5 + 4a;.
73. V3a; + 2-V2aj + l=V^Tl.
74. V2aj + 3-V8a; + 5 = -V4a;-l.
75. 2V3a; + 2-V6a;-3 = 3V3a;-l.
4
76
. V3a; + Va; — 2 =
77. V3a; + 1-V2a; =
^/x-2
5
78. V3a; + 3-Vaj-l =
V3a; + 1
2
79.
3V^qp4_V2a;-9 =
8
V2¥^^'
QUADRATIC EQUATIONS 119
3
80. V3a;-5+ ^_ = 2V^^=^.
■Vx — 1
81. Va;4-2 =
82.
83.
84.
+ Var^ + 7 'aj-V^+T
II. LITERAL QUADRATICS
Exercise 52
Solve:
1. 2a^-5ax = Sa^ 10. aar' 4-aa; + 2 =2a-a?.
2. 6aV-7aa; = 20. 11. 2a^ + a2 = a; + 3aa; + l.
3. Sx^-abx-2a^b'^ = 0. 12. ar'+aaj-2a;+l=2a2+a.
44. 5 aV -24 62 = 26 a6a;. 13. 2d'x'-a^x-9ax=a'-9.
5. 18 6V = 3 6ca; + 10 c^. 14. 6x'+ax+Sx=a'-{-a-2.
6. (B2-2aa; + a' = 4. 15. a V - a^ + 2 6a; = ft^.
7. o^-a^ + 6x + 9 = 0. 16. 6 V+a6a;-4 ar'=2 a(3 a;+a).
8. 4a^ = 4aa; — a^ + l. 17. a^a.'^— a;— aa:^— aa;=(a+l)^.
9. aa^ = 3a; + 4a-6. 18. 4:af-4.ax = b-a^ + c.
19. 4:a'x(x + l) + (a-l)(a + l) = 0.
20. a2a^-62 = aa;2 4-6aj. 22. 4a^-a2 = 2a + l.
21. a^a^ - (6 -1)=^ = 0. 23. 9aV-c2 + 6c = 9.
il^O QUADRATIC EQUATIONS
2^. 2aa^—bx = cx. ^^ 1 , 1 a + b
6o. \- = —
25. 3ax' + 4.bx + 5c=^0. « + ^ ^ + ^ ^^
26. aay^-\-2bx-\-3c = 0. ^q __j^_ = l_i^l.
„ ' x — a-\-b X a b
27. arH-pa; + g' = 0.
28. lx^-mx-lm = 0. 37. ?iL±_? + ^Lul^ = 2|.
2a-a; a4-2a; ^
29. a;2-2aaj-2a;+a2H-l=0.
30. (a2-6>2^a2(2.'c-l).
31. (l-a'){x-\-a)=2a(l-x')
30. (a2-6>2_a2(2.'c-l). 38. ^L_J_ 1^_ 2a = 0.
a
39.
a; a;— 1
1 111
32. - — bx = — a. a — x b — x a b
x — 1
o^ ^ .^ x — a 2a , 2x-\-3a
a; g — 1 _ a;H-2 . 18a -. _ x-{-a
a + 2 X ~x(a-\-2) ' 5a—x—Sb ~ x-^2b'
42. ax^-^^^^±^ + bx-^ = -^^-bx.
a — b a—b
43. ^ 1^ 3a^-2a ^^ _1- + -J l-=0.
a; (a — 1) (2 a — 1) a; + a x + b x -{- c
45. V2ar* — ax — a + 2 = a — 2.
46. V3 a^ — 4 «.« + 1 = 2 (a; — a).
47. Va; — a+V3a; — 2a=V2a; + 5a.
48. V^ + "^ a — Vaa; H- a^ = Va.
6 a
49. V^^-^ -^ = — Vx-b.
^x — a Va; — b
50. V6 + a; — V6 — x = V&.
51. 4a;(Va — «) = a — 6.
QUADRATIC EQUATIONS
52. Vx -\-a^-\- ^x — 2d^ = V3 x.
121
53. V3a^-4aa; + l=2(a-l).
54. ^/2a^x^ — 6ax — a^-\-5 = a — l.
55. Va; — a — Vcia; = Va; + a.
56. Vax —b — -Vax +b = ■\/arx -\-2ax — (ib.
III. EQUATIONS IN THE QUADRATIC FORM
Exercise 53
Find all of the values of x :
1. a5^-13a.-2H-36 = 0. 15. 3a;* + 4a;^ = 4.
2. 4a;^-29a;2 + 25 = 0.
3. 9a;^-28a^ + 3 = 0.
4. 9a;^+29aj2 = 80.
5. aj4-16 = 0.
6. 0^4-8 = 0.
7. a^-a; = 0.
8. 2«*-a^ = 15.
9. x^ — x = (}.
10. a;^ + 64a; = 0.
11. a;^ = 7a^ + 8.
12. 8a^ = 27.
13. a; + 4Va; = 5.
14. 2 a;^ — 5 a.'^ = 3.
16. 9a;3_37a;^ + 4 = 0.
17. 3a;4-5a;* = 12.
18. 12 a;^- 11 ^ = 15.
19. a;«-7ar^ = 8.
20. 4a;^-17A/^ + 4 = 0.
21. a;'' 4- 26^/^ = 27.
22. a;^ + a^ + 1 = 0.
23. a;-i + a;"^ - 6 = 0.
24. 2x-'-5x-'=:12.
25. Sx~^-^7x-^ = 6.
26. 2</x-
7 a5-3 = 4.
27. 9x-^ + 4 = 37-J/^l
28. 2.T~^-5v^^' = 3.
122 QUADRATIC EQUATIONS
29. 8-s/F^ = 15^^3 + 2. 36. V3^^-3^/3"^^ = 10.
30. 3x-^ = U-19VsF'\ 37. 2V5a; + l4-^5a;+l = 6.
31. 4a;-^ + 4a;-^ = 3. 38. ^2^31 4. 3 -^2 a; - 1 = 4.
32. (x + iy = 3(x + l)-\-A0, 39. 2(a; + l)*-3(a; + 1)^ = 2.
33. (x'+3xy-2(x'+3x)=S. 40. (4 » + 3)^-^/4^+3 = 6.
34. (a^-4a;)2-9(a^-4a;)=36. 41. 3 V3^+l + -</3^+l=14.
35. (a;-l)*-13(a;-l)24-36=0. 42. 2(2a;+3)-6V2a; + 3 = 3.
43. 2a^ + l-2V2ar^ 4-1 = 3.
44. ic2-a; + 4-6Va:^-x + 4 + 8 = 0.
45. (a^-xy-a^-{-x = 30.
46. 3ic2_4^^3y3ajj_4a._^2 = 2.
47. 2a? + 3x-\-V2a^-{-3x-\-7 = 5.
48. i»2 = 8-3a;-4Va^ + 3a;-3.
49. 3a^-aj = 6V3a^-aj-6 + 22.
50. a^ = 5x + 10-2V3f-5x-2.
51. 2a^ + a; + 5 = 5V2a;2 + a; + l.
52. (a^-x + iy = 3a^-3x-^l.
53. ic2-a; + 5V2a^-5aj + 6 = |(a;4-ll).
54. V2^+9^T9+V2^^+7¥+5 = V2.
55. aj(2a;-3)(2a^H-13aj + 20)=0.
56. ax(x-l)(x'-{-l)(x'-S) = 0.
57. (a^-x-12)(x'-hx-90)(a^ + x-110)=0.
SIMULTANEOUS QUADRATICS 123
SIMULTANEOUS QUADRATICS
Exercise 54
Solve :
1. 2y-Sx = 7', 3a^-4a^-42/2 = 15.
2. 0^ + 2/2 = 58; xy = 21.
3. 3a^-2a^ = 24; 5a^-4/ = 44.
4. a? — 2 y = 1 ; fl;2/ = 3-
5. x' + xy + f = lS', a^-xy-hf = T.
6. 2a;-2/ = 7; &a^-Sy^ = -7.
7. 3a^-52/2^28; 3 ajy - 4 2/^ ^ g^
8. 2a^ + a52/ — 2/^ = ^;^ + ^ — 2/^ = 1'
9. 3a; + 4?/ = 2; ar^ - oji/ - 5 1/^ = 1.
10. a;4-3y = -4; 6a^ + 13 a;2/-5/ = 21 2/- 12a? + 18.
11. a;H-2/ = 7; a^ + 2/^=29.
12. a^ + f=21S', x-{-y = 2.
13. a;2/ + 32/^ = 20; a^-3icy = -8.
14. a;4-32/ + 4 = 0; 2a^-52/^ = 5.
15. 2a;-32/ = 3; 4:0^-15-7 xy = 0.
16. a^-a^ + 2/^=21; 3^ + 2/^ = 189.
17. 3a^-52/2 = 7; 4:Xy-y^ = 7.
18. 6a;-82/ + 23 = 0; 32/^-5a^-2a; = 26.
19. a;-2/ = l; a^-2/3^i^
124 SIMULTANEOUS QUADRATICS
20. 2x' + 3xy-4.f- = 10; 7x~5y = 9.
21. ar^_3/ = l|5 2x2-j-/ = 4f.
22. 1 = ?. l_l = l
23. a;2_^^_j_^2^3^. ^_^^^^,^
24. (^ + 2/)'-5(x-}-?/)=36; 9a;-42/ = 29.
26. 3a^ + 22/ = 13; a^z/2_^y_30
27. 3a;2 + 52/- = 17; Aa^-3y'=lS.
28. 0^ + 2/2=62-3^-2/; a:?/=14.
29. 2a^-3aJ2/ + 4/=6; x'-^Sf = 7.
31. a^H2/'=626; x-^y = 6.
32. 2a;2 + a;y/_32/2 = 8; a;2_2^2_7
3^ 3x2-5/-3.T-22/ = 9.; 2x~3y = l.
34. 0.-^ + 2/5 = 1056; a; + 2/ = 6.
35. a;-i_2/-i = l; a;-3 _ 2/-3 ^ 3 j^
36. a; + a^2/ = 2; 2/ + a?2/ = 4i-.
37. 3x'-{-2xy-2y' = 6', 2x' + xy~3f = 3.
--38. 0^-2/2 = 16; a^ + 2aJ2/ = 4-2/2.
39. xy = l^- 2x-5y = 2.
40. a;2_|_^2^^^^^20; x + y = 3.
41. -1 L_ = _44. ? . 4_J^
«-2/ 0^4-2/ ^ ' x^y xy
SIMULTANEOUS QUADRATICS 125
42. --^ = -2|; 2x-^3y = 2.
y X
43. 2x + y + 2xy = 5; x-\-3y + 2xy = l,
44. 2x + 3y = 10; 23^f + ^5xy = 72.
45. a?2_^X2/ + / = '^; ^-^2/ + / = 19.
46. 5a^-2/' = ll; 3a5?/ + / = -9.
47. a^/ + 14a;2/ + 24 = 0; Sx + y = 5.
48. a; 4-2/ = 4; aj* + 2/^ = 82.
49. x^ + f = S7) x'y-{-xy' = -12.
50. a^ + i/2-5aj + 52/ = 30; a;2/ = 8.
51. x' + xy + f = 19; x' + xV -]- y' = 931.
1 1 ^ 1 I 1 fti
52. i_± = 4; - + - = 8|.
a; 2/ aj2 2/
53. x-V^ + 2/ = 9; ^2 _^ 0^2/ + 2/' = 18^-
54. a^4-2a^2/ + 32/' = ti' + 262; x + 2/ = «-
55. o^ + y^-{-x-y = 32', xy = 10.
56. 0^*4-2/^ = 5; a;*2/^ = 6.
57. x^y=-117', Vx+-y/y = 3.
58. aj^-22/^ = l; x-Sy = 19.
59. x2 4-a;2/4-2/' = '^5 a^' + «'y + 2/' = 91.-
60. a; 4- 2/ = 3; x^ + y^ = 33.
61. a;2 4-42/'-x-22/-42 = 0; a;2/ = 12.
e2. a^-f = m', x^y-xy^ = 30.
63. a;-2/ = 2; Vac4-V2/ = 2.
64. x + y = 13 + V^) a^^f = 273-xy.
126 PBOPERTIES OF QUADRATICS
65. J^+2Jl = 3; x + y = 5.
^y ^x
66. 2a; + Vi^=12; 2/4-V^=l8.
en, x^ + xy-\-y'^ = 3? — xy-\-y'^ = l.
68. x^ -\- y^ =^ xy = 1. 69. or — y"^ — xy = x-{-y.
IQ. x^ + y'^ = Zxy-l; x^-^y^ = ll.
71. ^ + ^ = _26: 1^ + 51^=1.
2a 6 X y
72. (a;-22/)2-a; + 22/ = 6; 3a;-52/ = ll.
73. a;y + a; + 2/ = 7; o^ (a; + 2/) = 12.
74. 2a:2_^3y2^8. 2(a;- l)2 + 3(2/ + l)2 = 5.
76. x^^^/'^^aj^/^ig. x-\-y = xy-7,
PROPERTIES OF QUADRATICS
Exercise 55
Form the quadratic equations which will have the following
roots ;
1.
7,5.
7.
0, 5.
2.
2, ^.
8.
-2i, 0.
3.
6,-4.
9.
a,-l.
4.
4, -If
10.
7,-7.
5.
if
11.
V5, -V5.
6.
-1, i-
12.
V-3, -V
3.
13.
a, a — 1.
14.
3+a, -3-2a.
15.
2.1-3«.
16.
±10.
17.
±\/a-l.
18.
1 + V2, 1-V2.
PROPERTIES OF QUADRATICS
127
19. 3±V2.
20. ±Vll-5.
7±V70
21
22.
2
3±V3
23.
24.
-7± V5
a± Va'-l
25. 5±V-1.
26.
27.
28.
29.
30.
5± V^r2
-7±2V-1
2
5±3V^
Without finding the actual values of x, tell what the sum of
the roots is ; their product ; their character :
31. x^-5x-24. = 0.
32. a^ + 5a;-l = 0.
33. 2a^-3a;-f 1 = 0.
34. 3a^H-«-10 = 0.
35. x^-Sx-^5 = 0.
36. 5a^-6x-\-2 = 0.
37. 4a^+4a; + l = 0.
38. 4ar2 = _a;-f4.
39. 9x^ + 1 = 6x.
40. 12af-\-7x = -6.
41. 4a^-3a; = 0.
42. 4a^ = 7.
43. x'^-x = l.
44. 3x^-i-5x-}-3 = 0.
45. 25ar^ = 10a;-l.
46. 3a^4-5a; = 0.
47. 3a^ + 5 = 0.
48. 2a^-a; = l.
49. 16a^-40a; = -25.
50. 7a^ + 13ic = 5.
Find the values of k which will make the following equations
have equal roots :
51. 2x^-2x-\-k = 0.
52. ko(^-4:X-\-S = 0.
53. x^-\-x = — k.
54. kx^=:3x-2.
55. Sx^ + 2x = l-k.
56. A;a^-A:a; + 1 = 0.
128 PBOPERTIES OF QUADRATICS
57. 5a^ = 4a;-2A;4-l. 61. 4.x' = kx-k-5.
58. a:^-kx-^9 = 0. 62. lla:^-\-l = 3x-kx^-\-kx.
59. kx^-i-kx = -Sx-9. 63. ka^-kx=7 x^-j-9 x-25.
60. ic2 + 49 = A'x + 3a;. 64. 3feic2+6A;=5a;(A;4-3)-7.
Resolve into factors :
65. a^-3a; + l. 68. a^ + 4. 71. a^-Saft + fe^.
66. x^-x-3. 69. a^ + ic + l. 72. 17-8a; + a^.
67. Sx^-2x-2. 70. x^-lxy-y\ 73. 5a^ + 8a;-2.
74. Explain the rules for determining whether the roots of
an equation are real or imaginary. Equal or unequal. Rational
or irrational.
75. If the sum of the roots of a quadratic is 3 and their
product is 2, find the difference of the roots. Find the differ-
ence of the squares of the roots. Find the sum of the recipro-
cals of the roots.
76. Find the condition that one root of ax^ -\-'bx-\-c =
shall be the reciprocal of the other. Find the condition that
one root shall be double the other. One three times the other.
77. If m and n stand for the roots of 2 ic^ -f 5 .'c — 3 = 0, find
the values of : (a) m + n. (c) m — n. (e) — |
^ m n
(b) mn. (d) m^ — n\ (/) m^-\-7i\
78. Find the values of the same expressions in the equation
3 a:^ = 13 a; + 10. Also in equation 3 .t^ — a^ + 1 = 0.
79. Form the quadratic whose roots shall be | and |. Form
that whose roots shall be | and |. Compare the results.
hatio and propohtion 129
RATIO AND PROPORTION
Exercise 56
1. Find a mean proportional between 5| and 27. Between
m and n^. Between -— and ' ^ »
a a
2. Find a fourth proportional to 3, 5, 12. To a, a + 1, a^.
To 6, 8, lOf To 8, lOi 6.
3. Find a third proportional to 4 and 10. To 3 and 3|.
To a and a; - 1. To i and |.
4. Solve 2a;-l:3a;-2 = 3(a; + l):5a; + l.
5. Solve a;-5:3a;-fl = 5-8a;:3(l-2a;).
6. Solve l:l = -l_:i.
a 6 a—b or
7. Solve — : -— ! — - = x:a-\-
c{a-\-c) a^ — c^ a — c
8. Solve
2x^-3x + l:3x'-3x-^l = Say'-2x-5:4:x'-2x-5.
2a^-4:X-l x'-\-x-2
9. Solve
2a^-f2a;-l a^+13a;-2
10. Solve .^ + 3a.-7^a^ + 4a. + 10,
a^ — 5a;4-6 a:^ — 4a; + 4
11. Solve ^^-^^-^==^ + ^ + ^.
12. Solve ^-2.^-4-2 2^ + . ^1
a^_3ar^ + 2 2x2-x-l
13. Solve V« + 4 : Va; - 1 = V6 a; + 6 : V5 a; — 9.
14. Solve V3a;-2:V4a; + l = V7a; + 2:2V5aj-l.
R. & S. EX. IN ALG. — 9
130 RATIO AND PROPORTION
15. Two numbers are in the ratio* of 3 : 7 and their sum
is 60. Find them.
16. Three numbers are in the ratio of 2 : 3 : 4 and their sum
is 63. Find them.
17. Find two numbers in the ratio of 2 : 5, the sum of whose
squares is 464.
18. Find three numbers in the ratio of 1:2:3, the sum of
whose squares is 126.
19. What number added to each of the numbers 2, 5, 11, 15
will make the sums proportional ? •
20. Find a mean proportional and a third proportional to 5
and 20. Also to 3i and H.
21. It a: b = c:d, prove the property of " composition '^ by
use of the equivalents, a = bx and c = dx. Prove " division "
by the same method.
22. If a:b = c:d = e :f=g : h, prove by the method of
example 21 that a-\-c-\-e + g'.h-\-d 4-/4- h = a:b=c: d = etc.
23. If a: b = c: d, prove that a4-3c: b-^Sd = 2 a-\-c: 2b-\-d.
24. If m : n=p : q, prove that m-^n : p-\-q = m — 2 n : p—2 q.
25. It X : y = z : w, prove that
x^ -\- y^ : z^ + iv^ = (x — nyy : (z — 7iwf.
26. It p : q = r : s, prove that
Vi>^ 4- 7^ : Vg^ -\-s^ = ap — br:aq — bs=p:q.
27. It a:b: :b :c, prove that a-{-Sb:b-\-Sc = a:b by use
of the equivalents a = cx^ and b = ex.
28. If 2/ is a mean proportional between x and Zj prove that
x-2y:y-2z = 2x-'^y\2y-^z.
RATIO AND PROPORTION 131
29. If a, b, c, d are in continued proportion, prove by use
of the equivalents a = da?, b = daf, G = dx that a + 6 + c : a + &
= 6 + cH-cZ:54-c.
30. If a, b, c, d are in continued proportion, prove that
a-\-b^:c-\-c^ = b-{-c':d-\-d\
31. If a, b, c are in continued proportion, prove that
a + b :b + c = b^: ac^.
32. If a, b, c, d are in continued proportion, prove that
a^^b' -{-(?: b' + c'-\-d'={a + c){a-c):(b-\-d){b-d) = a':b^
33. If _^_ = _L=_!_, prove that x-y-{-z = 0.
b + c a-\-c a — b
34. If _l_=_!^ = _i!_, prove that l + m + n = 0.
b — c c — a a — b
35. If a + 2b-\-c:b-\-c = a-{-b:b, prove that 6 is a mean
proportional between a and c.
36. Find two numbers in the ratio of 2 : 3 such that the
sum of their squares is to their product increased by 2, as 2 : 1.
37. If 1 be added to each of two numbers, their ratio is 1 : 2.
The difference of their squares is to 3 more than their product
as 5:3. Find them.
38. There are two numbers such that the ratio of the sum
of their cubes and the cube of their sum is 7 : 1 ; and if 6 be
added to each, the ratio of these sums is 1 : 4. Find them.
39. For what value of x will 2 a; — 1 be a mean proportional
between x-\-5 and 4 a; — 13 ?
40. What values must x have in order that 2 a; — 7, 3 a; + 1,
4 a; — 3, 5 (a; + 1) may form a true proportion ?
132 VARIATION
VARIATION
Exercise 57
1. li X varies as y and y = 2 when x = 12, find x when y = ^.
2. It xccy- and x = ^ when y = ^, find ?/ when a; = 18.
3. If A varies inversely as B and A = — 6 when ^ = — i find
^ when -B = |.
4. If ^ varies jointly as B and C and ^ = 9 when 5 = = 6,
find A when B = 5 and O = — 8.
5. If 07 varies directly as y and inversely as z, and a; = 2
when y = 3 and 2 = 6, find «/ when x = S and 2; = — 3.
6. li xccy and a; is 3 when y = -|, find an equation between
X and y.
7. If ic X - and ?/ = — 5 when a; = 2, find the equation joining
X and ?/.
2/
8. If a; X - and a; = 15 when y = 5 and 2; = 4, find a; in terms
z
of 2/ when 2 is — 1.
9. If a! X (2y + 5) and a; = 3 when ?/ = — 2, find y if x = 6.
10. Given that ?/- x (a;^ + 1) and a; = 7, when j^ = 10, find x
when 2/ = VTO-
11. If u is equal to the sum of two quantities, one of which
varies as x and the other inversely as x, and if u = — l when
a; = |, and w = 1 when a; = 1, find the equation between u
and X.
12. If V is equal to the sura of two quantities, one of which
varies as a^ and the other inversely as 1/, and v = — 1, when
X =^, y = 2', and v = 7 when x = 2, y = 3 ; find the equation
for V in terms of x when ?/ = — 1.
VABIATION 133
13. Given that y = the sum of three quantities which vary-
as X, x"^, and x^ respectively. When x=l, ?/=4; when x=2,
y = S\ when x = 3, y = IS. Express y in terms of x.
14. If y varies inversely as ar^ — 1 and y = — 5 when a; = 4,
find X when y = — 15.
15. If y varies inversely as (2 x + 1) (x — 3) and y = — \
when a; = 2, find a; when y — 1\.
16. If the area of a circle varies as the square of its radius,
and the area of a circle whose radius is 7 is 154, find the area
of the circle whose radius is 10.
17. Find the radius of the circle equivalent to the sum of
two circles whose radii are 5 and 12 respectively.
18. The pressure of the wind upon a plane surface varies
jointly as the area of the surface and the square of the wind's
velocity. The pressure on a square foot is 1 pound when the
wind is blowing at the rate of 15 miles per hour. Find the
velocity of the wind when the pressure on a square yard is
36 pounds.
19. If w varies as the sum of x, y, and z, and tv = 3 when a; = 3,
y = — 4:,z=6, find xiiw = — 3,y = 3^,z = — 9.
20. If w is equal to the sum of two quantities, one of which
varies as x, and the other jointly as y and z, and w = — 3 when
a; = 2, 2/ = 6, 2 = — 1; and w = — 2 when a; = 4, ?/ = 2, 2; = — 3 ;
find the equation combining the four quantities, w, x, y, and z.
21. If the square of x varies as the cube of y, and a; = 3
when y = 2, find y when x = 24.
22. The area of a triangle varies jointly as its base and
altitude. Find the altitude of a triangle whose base is 23,
equivalent to the sum of two triangles whose bases are 15 and
22 and whose altitudes are 10 and 12 respectively.
134 ARITHMETICAL PROGRESSION
ARITHMETICAL PROGRESSION
Hzercise 58
In the following 16 examples tell what a is, what d is, what
n is. Also find I and s in each.
1. 5, 7, 9, •••, to 15 terms.
2. 6, 9, 12, ..., to 10 terms.
3. — 2, — 31 — 5, •••, to 45 terms.
4. 3, 3.1, 3.2, ..., to 300 terms.
5. 8, 7.5, 7, •••, to 60 terms.
6. 2|, 2^,21, ..., to55terms.
7. - 3^, - 2f, - 21 .-., to 75 terms.
8. 1 + a;, 1 + 3 a;, 1 + 5 X, • • •, to 10 terms.
9. Odd numbers to 37 terms.
10. Numbers divisible by 7 to 15 terms.
11. Numbers divisible by 3 to 20 terms.
12. 5, 10, 15, •••, to r terms.
13. 1, 2, 3, 4, •••, to X terms.
14. 2, 6, 10, 14, ..., to w terms. .
15. The first n odd numbers.
16. The first 2 71 even numbers.
Insert, between
17. 11 and 32, 5 arithmetical means.
18. 7^ and 30, 9 arithmetical means.
ARITHMETICAL PROGRESSION 135
19. 38| aud — 44|, 99 arithmetical means.
20. 17 and 3, 12 arithmetical means.
Find d and s if :
21. a = 5, Z = 25, n = ll. 23. a = 4, Z = 36, n = 24.
22. a = -13, Z = 26, n = 14. 24. a = 12i, ^ = - 13|, n = 40.
Find n and s if :
25. a = 6,d = 2,lz=S0. 27. a = 3J, c? = J, ; = lOf
26. a = -17, d = 4, ^ = 39. 28. a=9i, d = -i, l = -W^.
Find a and s if :
29. d = 3, ^ = 38, n = ll. 31. d=-2,l=-25,n = 27.
30. (7 = 1|, Z = 69, n = 41. 32. d = -|, Z = 6^, n = 20.
Find Z and d if :
33. a = 5, n = 9, s = 297. 34. a = 3J, n = 15, s = 78|.
35. a = -l|, n = 30, s = 530.
Find n and d if :
36. a = 8, ^ = 41, s = 294. 38. a = 8, Z = 0, s= 100.
37. a = 3i Z = 42|, s = 621. 39. a=-3^,Z=-36,s=-790.
Find a and /, if :
40. d = S, n = 13, s = 260. 41. d = i, n = 20, s = 102|,
42. d = -f, n = 8o, s = -306i.
Find a and d, if :
43. Z = 47, n = 23, s = 575. 44. Z = ll|, n = 37, s = 209J.
45. / = -16^, n = 43, s = 43.
136 ARITHMETICAL PROGRESSION
Find n and ?, if :
46. a = S, d = 2, s = 80. 47. a = 2, d = -S, s = -328.
48. a = o, d = — ^, s = 27.
Find n and a, if :
49. d = 5, l = S2, 5 = 119. 51. d = l, 1 = 6, s = 45.
50. d = -^, / = 5i s = 2o. 52. fZ = -|, / = -3, s = 13.
53. How many numbers are there between 100 and 1000
that are exactly divisible by 7 ? Find their sum.
54. Find the sum of all the numbers of two figures each
that are divisible by 8.
55. Find the sum of the first 50 odd numbers.
56. In the series 2, 5, 8, •••, which term is 98 ?
57. How many terms must be taken from the series 3, 5,
7, •••, to make a total of 255 ?
58. Which term of the series li 2, 21 •••, is 24? How
many consecutive terms must be taken from this series to
make 84?
59. The 7th term of an A. P. is 17, and the 12th term is 27.
Find the 1st term. The 3d term.
60. The 10th term of an A. P. is |, and the J 8th is 3f .
Find the 1st term. The 100th term. Sum of 20 terms.
61. How is a single arithmetical mean between 2 numbers
found most readily ? How do you determine whether or not
3 numbers are in A. P. ?
62. Find x, so that 3 — 5x,l-\-2x,4:-\-7x, shall form an A. P.
63. The sum of 4 numbers in A. P. is 46, and the product of
the 2d and 3d is 130. Find them.
64. The sum of 3 numbers in A. P. is 27, and the sum of
their squares is 275. Find them.
ARITHMETICAL PROGRESSION 137
65. A body freely falling from a position of rest will fall
16 J^ feet the first second, 48^ feet the second second, SOy^ feet
the third, and so on. Find the distance fallen during the 10th
second. How far in 10 seconds? How far in 20th second?
How far in 20 seconds ?
66. Find x, so that S -{- 2 x^ 5 -{- 6 x,9 + 5 x, shall form an A. P.
67. Which term of the series 2^, 3f, 5, •••, is 45 ?
68. How many consecutive terms in the series 2^, 3|, 5, •••,
will make 67^ ? Interpret the negative result.
69. If the 6th term of an A. P. is 9 and the 16th term is
22J, find the 25th term and the sum of 30 terms.
70. Find the sum of the series x, Sx, 5x, 7 x, ••., to x terms.
71. Find the sum of all the numbers between 100 and 600
that are divisible by 11.
72. Find x, so that 2a; — 1, 3a; + 2, 6aj + 8, shall be an A. P.
73. What will x and y each be, if the four terms 2x — y,
x-\-2y,3x + y^7x — 10, form an A. P. ?
74. Find the sum of 15 terms of an A. P. of which the
middle one is lOJ.
75. Find the sum of '1±1 -f- !?i±^ + ^?i±^ . . . to n terms.
n n n
76. A boy travels at the rate of 1 mile the first day, 2 the
second, 3 the third, and so on; 6 days later a man sets out
from the same place to overtake him, traveling 15 miles every
day. How many days must elapse after the second starts
before they are together? Interpret both results.
77. The sum of n terms of the series 21, 18, 15, •••, is equal
to the sum of the same number of terms of the series 3, 3^,
3_6_, .... Find n.
78. Find the sum of 41 terms of an A. P. whose 21st term is
100.
138 GEOMETRICAL PROGRESSION
GEOMETRICAL PROGRESSION
Exercise 59
Find I and s in each :
1. 3, 6, 12, •.., to 8 terms.
2. 2, 8, 32, ..., to 5 terms.
3. 40, 20, 10, ..., to 6 terms.
4. 2.1, 21, 210, ..., to 5 terms.
5. 54, 18, 6, ..., to 5 terms.
6. 3.2, 0.32, .032, ..., to 6 terms.
'^' ^j f? \h •••? to 5 terms.
8. I, 4^2, ..., to 7 terms.
9. 11 —3, 6, ..., to 9 terms.
10. - 5, 15, - 45, . . ., to 5 terms.
11. 34, If, I,..., to 10 terms.
12. 16J, -111 71 ..., to 5 terms.
13. l+a; + «2 + a^---, to6 terms.
14. 32-16 + 8-4 + 2-1..., toTi terms.
Find r and s, if :
15. a = 3, ^ = 48, w = 5. 16. a=^n, 1 = 4.05, n = 5
17. a = 131 Z = 17, ^::=a
Find a and s, if :
18. Z=i, 71 = 6, r=i-. 19. / = 85i n = 5, r = lj.
20. Z = |, 71 = 5, r = -2
GEOMETBICAL PROGRESSION 139
Find n and s, if :
21. a = 5, 1 = 160, r = 2. 23. a = 24, /=|, r = f
22. a=3, Z = 1875, r = 5. 24. a = f, Z = -24, r = -2.
Find r and r?., if :
25. a = 2, Z = 486,s = 728. 27. a = 1|, Z = 135, s= 201f.
26. a = 56, Z = lf, 8 = 1101 28. a = |, Z = - ^^^ ^ s = - 8|f f .
Insert, between
29. 4 and 972, 4 geometrical means.
30. 7 and 896, 6 geometrical means.
31. 5^ and 40 J, 4 geometrical means.
32. 20and— yl-g^, 8 geometrical means.
33. 7^ and ff, 4 geometrical means.
Find the sum of each series to infinity :
34. 6,3, H,.... 38. 8|, -6|,5,-...
35. 1, -|, 1 .... 39. 8.3, 0.83, .083, ....
36. 15, 5, If, .... 40. .72, .0072, .000072, ....
37. 18,12,8,.... 41. 1^,0.75,0.5....
42. 0.4545, .... 44. 3.8181, .... 46. 2.34848, ....
43. 0.05454,.... 45. 5.12727,.... 47. 1.026363,....
48. If the 3d term of a G. P. is 36 and the 6th term is 972,
find the 1st and 2d terms.
49. If the 4th term is 24 and the 8th term is 384, find the
first 2 terms.
50. The 3d term is 4 and the 7th is 20^. Find the first
2 terms.
140 GEOMETRICAL PROGRESSION
51. In the G. P. 2, 6, 18, .••, which term is 486 ?
52. How many terms must be taken from the series 9, 18,
36, •••, to make a sum of 567 ?
53. How many consecutive terms in the series 48, 24, 12, •••,
are required to make 95 J ?
54. The 1st term of a G. P. is 8. Its sum to infinity is 32.
Find the ratio.
55. How can a single geometric mean be determined most
readily ? How does one test a series to determine whether it
is a G. P. or not ?
56. Find ic, if 2 a; — 4, 5 a; — 7, 10 a; + 4, are in G. P.
57. There are 3 numbers in A. P. whose common difference
is 4. If 2, 3, 9, be added to them respectively, the sums form
a G. P. Find the numbers.
58. The sum of a G. P. to infinity is 18 and the 2d term is 4.
Find the 1st term and ratio.
59. If the series a; -f 1, x-\-S,4:X — 3, is geometric, find x.
Find a; if it is an A. P. Find the 4th term of the series in
each case.
60. Tell whether each of the following series is arithmetical
or geometrical :
(a) 3, 6, 12, .... (c) 12, 18, 25, ....
(6) 6, 12, 18, .... (d) 3i, H, 0.6, ....
61. The sum of three numbers in G. P. is 65. The sum of
the first two is i the sum of the last two. Find them.
62. Divide 49 into 3 parts in G. P. such that the sum of
the 1st and 3d parts is 2i times the middle part.
63. The sum of 3 numbers in G. P. is 14 and the sum of
their reciprocals is f . Find them.
GEOMETRICAL PROGRESSION 141
64. Insert between 6 and 16 two numbers, such that the
first three of the four shall be in A. P. and the last three
in G. P.
65. If the series 3^, 2|^, •••, be an A. P., find the 105th term.
If a G. P.J find the sum to infinity.
66. The sum of $ 240 was divided among 4 men in such^a
way that the shares were in G. P., and the difference between
the greatest and least shares is to the difference between the
other two, as 13 : 3. Find each share.
67. What number added to each of the numbers 2, 5, 11,
will make sums that are in G. P. ?
68. Find x, so that 5-\-x, 5 — x, 2(1 — 5 a;), shall be in G. P.
69. If 4 a? — 1, 6 a; + 1 , 5(2 x -f- 1), are in G. P., find x and find
the ratio. Also find the next term.
70. If the first term of a G. P. is 6 and the sum to infinity
is 18, find the third term.
71. If a man ascends a mountain at the rate of 81 yards the
first hour, 54 yards the second, 36 yards the third, etc., how
many hours will he require to ascend 211 yards ?
72. There are 4 numbers, the first three of which are in
G. P., and the last three are in A. P. The sum of the first and
last is 14, and the sum of the second and third is 12. Find
the numbers.
73. A ball thrown vertically into the air 150 feet falls and
rebounds 60 feet. It falls again and rebounds 24 feet, and so on
until it comes to rest on the ground. Find the entire distance
through which the ball has traveled.
74. Prove that equimultiples of a G. P. are also in G. P.,
and that alternate terms of a G. P. form another G. P.
142 PERMUTATIONS AND COMBINATIONS
PERMUTATIONS AND COMBINATIONS
Exercise 60
1. How many numbers of 5 different figures each can be
formed from our 9 significant digits ?
2. How many words of 4 letters each can be formed
from the 26 letters in our alphabet, no letter being repeated
in the same word ?
3. Find the number of committees, each containing 5 boys,
that can be selected from a room of 20 boys.
4. Find the number of combinations of 8 objects each that
can be formed from 25 objects.
5. How many different words can be formed from the
letters in the word TJiursday, using all its letters each time ?
6. From the members of a party of 30 people, a board of
4 officers is to be chosen. In how many ways can this occur ?
7. From the letters in the word Repiihlican how many
words of 4 letters can be found ? Of 5 letters ? Of 7 letters ?
8. The prime factors of a certain number are 2, 5, 7, 11,
and 17. How many exact divisors (except itself and unity)
has this number ?
9. It is required to place 20 dissimilar bouquets in the form
of an arch. In how many ways can they be arranged ?
10. From the 9 significant digits, how many numbers can
be formed each containing 1 digit ? Two different digits ?
3? 4? 5? 6? 7? 8? 9? All together?
11. There are 25 points in a certain plane; these are joined
so as to form triangles having the vertices at the points. How
many triangles will there be ?
PERMUTATIONS AND COMBINATIONS 143
12. From the letters in the word handiwork how many
words of 5 letters can be formed ? How many of these will
contain the h ? the w ? How many will begin with d ? How
many will contain both d and i? How many will not con-
tain d ?
13. A man has 5 pairs of trousers, 8 vests, and 6 coats. In
how many different costumes can he appear ?
14. Six persons enter a car in which there are 10 seats. In
how many ways can they be seated ?
15. In how many ways can a baseball nine be arranged
provided the pitcher is always the same ? Provided the pitcher
and catcher are always the same individuals ?
16. In how many ways can 10 people arrange themselves
around a circular table ?
17. How many words can be formed from the letters in the
word latin, the 2d and 4th being always vowels ?
18. How many words can be formed from the letters in the
word united, the even places being always occupied by con-
sonants ?
19. How many words can be formed from the letters in the
word education, provided the 2d, 4th, 6th, and last letters are
always consonants ?
20. From our 9 digits how many numbers can be formed,
each containing 6 figures ? How many of these will contain
the figure 5 ? How many will not contain a 7 ? How many
will contain both 5 and 7 ? How many will begin with 3 ?
End with 4 ? How many will be odd ?
21. From 10 gentlemen and 8 ladies how many committees
can be chosen, each containing 3 gentlemen and 2 ladies ?
22. From 10 consonants and 5 vowels how many words, each
containing 4 consonants and 3 vowels, can be formed ?
144 PERMUTATIONS AND COMBINATIONS
23. There are 8 Democrats and 10 Kepublicans belonging to
a certain board. How many committees can be chosen each
having 4 Democrats and 5 Kepublicans ?
24. Out of 4 vowels and 9 consonants there are words to be
constructed, each consisting of 2 vowels and 6 consonants.
How many can there be ?
25. From 6 white balls, 4 red balls, and 8 black balls, how
many combinations can be made each to contain 3 white, 2 red,
and 4 black balls ?
26. From 4 labials, 6 vowels, 5 palatals, how many words
can be made each consisting of 2 labials, 3 vow^els, and 2
palatals ?
27. How many different sums of money can be made from
the following coins: cent, 5-cent, dime, quarter, half dollar,
and dollar ?
28. A guard of 5 men must be selected every night out of a
detachment of 32 men. For how many nights can a different
guard be selected ? How many times will each soldier serve ?
29. A company of 15 merchants, 12 lawyers, and 8 teachers
wish to form a commission from their number, consisting of 4
merchants, 3 lawyers, and 2 teachers. How many ways are
there in which they can do it ?
30. Find the number of permutations that can be made from
the letters in the following words using all the letters :
(a) Recess. (c) Bumblebee. (e) Concnn-ence.
(b) Possess. (d) Tennessee. (/) Unostentatious.
31. In how many different ways can one mail 4 letters in a
village containing 7 letter boxes ?
32. How many different quantities can be weighed with the
following weights: 1 ounce, 3 ounces, 8 ounces, 10 ounces,
1 pound, 5 pounds, and 10 pounds?
BINOMIAL THEOHEM 145
33. With 2 violet, 2 indigo, 3 blue, 4 green, 1 yellow, 1 orange,
and 2 red flags, how many signals can be made if all the flags
are used and always kept in a vertical column ?
34. Prom 7 consonants and 5 vowels how many words can
be formed, each consisting of 4 consonants and 2 vowels ?
35. A plane is determined by 3 points, if they are not all
in a straight line. How many planes are determined by 100
points (no four of them lying in the same plane) ?
BINOMIAL THEOREM
Exercise 61
Expand :
1. (a-2y. 6. (2m-\-ny. ' 9. (■</6-2)«.
2. (2a + l)^ 6. (3a-Va6)*. 10. (V2-h2V6)3.
3. {l-^a'xy. 7. {a-V^-iy. 11. (Va-Vw)'*'.
4. (1-2/y. 8. (V2-V3)^ 12. (a^-^a^c/.
13. (1-3V3)1 20. r^ + ^Y-
14. (Vi-3V2^)*.
21. {a</x-'-x^a-y
22. (2aV2^+-v/4)*.
15. (V3«-^2a;2/)^
16. (2-V^^y. ,_ ,,
^ ^ 23. (V2a-</-3ay,
17. (3V-1 + 2V5)'. r2Va^3V2l^
24. ^ -zzz -\ — —
18. {cix~'^ — •\/a~'^x) • I Vn^ ^a
^^' V^-3V^=3^ . 25. -^-^^
V ^ .S « / la V a V c J
R. & S. EX, IN ALG. — 10
146 BINOMIAL THEOREM
Find only the term required :
26. TheTth termof (aj + V^)".
27. The 5th term of (1 - 2 xf.
28. The 4th term of (a + 3 ■\/xf\
29. The 6th term of (2 n ^/m - 1)".
30. The 8th term of (a V^ - a; VaP.
31 . The 7th term of (i - 2 a; ^^)".
32. The 5th term of (Vox +^?y«.
33. The4thtermof (V6- V3)«.
34. TheSth termof (V3 + V-2)».
35. The 7th term of ( V2 - a V3xy\
36. The 9th term of (J^ ^a + 2 V3^)^.
37. The 4th term of ( V2 - 5 V^^.
88. The 5th term of [^ + -^^^T-
39. The term containing o^ in (1 —xVxy^.
40. The term containing x' in (a;-f 2V^)^.
41. The term containing x^^ in (■\/x—-^xyy\
42. The term containing x^ in (2 \/x— V2xy.
43. The term containing x-* in f— ^—\ .
V6a: V^V
BINOMIAL THEOREM 147
Expand to four terms :
-I .. 1
44. (a + 6)-^ 52. (x-2-^/ax^-^, 59
3. ^ ' Va-\-2¥
45. (a-a;)-\ 53. Va^ + Va. o ah
60.
46. (a;-2)-^ 54. (l + 5f(2^x)i ^a'^ - aft^
47. (a + 2/)i 55. (a6«-36V«)l ^^^ T^^^'
48. (a2_4)-2^ 56. 5(x-^-\/2xy)-K gg. (a2_3V2)l
49. (l-2^-x)-\ 57. C-L_y3^y'^ 63. -—4
Wa J VaVx
50. (aar« + l)^.
58. 64
ax
2y
51. (b^-2by\ a-b^/c (x-2^xyy
Find only the terra required :
65. The 3d term of (a + b)'\
66. The 4th term of (x — y)-\
67. The 4th term of (1 +3Va5)i
68. The 5th term of (a;- 2 VaJ^) -2.
69. The 5th term of {l-^/2x)-\
70. The 6th term of (a + V-2«)-^
71 . The 6th term of (x^y + 3 yVx)^.
72. The 5th term of (x^ - 3 x-\/y)-\
73. The 7th term of \^=.-
a —'-wax
74. The 8th term of - ^^
Vl-4a26
148 LOGAmTBMS
75. The 5th term of '^{a-Q,^cy.
76. The 10th term of —
77. The 9th term of (V2 - 3 V^^)-\
78. The 8th term of (2 + V^^)~^.
79. The term containing a^ in (1 -\-x)~^.
80. The term containing ic" in (x'^ — 2 x)~\
81. The term containing x~^ in (v^ — 4Va)^.
82. The term containing a;"'^^ in T^^- ?:^T .
LOGARITHMS
Rsercise 62
Compute by four-place logarithm tables the values of the
following :
1. 55x3.86. 7. 823x756-- 4320.
2. 7.81 X 9.3 X .568. ^ 7.61 x 53.8 x 4.113
3. 8.52 X .0917 X 63.4. * 27.5x1.884
4. .097x63.8x51.14. _ 328x57.42
<7.
5. 8.76 X 95.32 -- 614.3. 134.2x3.876
6. 71.3x5.888-43.18. 10. 123.5x3.586-976.3.
11. 36.95 X 438.7 - (356.7 X 81.44).
87.63 X 563.8 x .0075 91.76 x. 00385x2.1 176
27.51 X 9832 x .0953 ' * 7.143 x .08885 x 11.58 '
LOGARITHMS
149
14.
75 X .03896 x .4427
83 X .008114 X 7.003"
15.
876.4 X 3.175 x .6511
8.465 x .1973 x 598.6*
16.
17. ■v^^9?f6.
18. V17 X 29.
19. ^/365.4.
20. V.0837.
29
5 X 78.3
X19.7
30
31.
'■i
9.02 X 1.762
3.117 X .0585'
5/ 17.44 X .(
\zt2 11 V 8
.0832
'42.11x8.104*
7.663 X 85.12 x .00681
43.27 X 95.16 x .007194*
21. -v/^00302. 25. -J/7.13 x 41.2.
22. -v/.00075. 26. V10.3^.049.
23. V93 X 2.78. 27. ^361^5.88.
24. ^951 X .037. 28. ^8.95 -f- 16.44.
3^ 853.4 X V:004176
627.1 X ■</ .06329 *
-. ^5/ 57.18 X 3. 876
^^' \7.116x.0
39
X .0485
7/ 5.192 X V63l8
^ 81.32 X .0638 *
32
■41
382 X 763.5
.03871 X 8124
33.
34,
35.
36.
4
03765 X 1.448
37.12 X 5.718
.4138 X VaiTr
3.108 X i/Tmi
5.167 X V38:27
77.38 X ^:09034
.7563 X -^2:087
.5432 X </M15
40
41
3/ .07162 X .00328 ,
^1.586 XA/37f77
^/ .0913 X \/:07652
' ^ 1.517 X 7.038
42.
43.
V
3/1.716 X 873.5
.0173xV3967
V
750 x (.83)
97 X (.0361)*
44
3 / 92.3
X .08763
.003151 X 5V30
150 LOGARITHMS
45 J 9.3I0 X -5^7^ 4g 37.5 6 /440V:0074
^10.14 xV:3876 * 583^ 19^600
47. V8.5I6 X v^9L763 x \/l998.
48. \/53:34xa/7JI6-V98J[5.
49. V8.176 X VMM X VTlI^.
50. -v^l.Tie X 8.513 X VtKM.
61. V73.14X 80.37 -^5768.
52. ^53.11 X V7:852 - 3 VTT^.
63. •v'3V85 X 2^916 x 5^45.
g4 J 876.3 X 5.173 x V.()08886
6.385 X 711.5 X v':0l776 *
gg J .07138 X V.00885 x ^1:762
.08195 X V.00176 x VsM2
gg^ J 57I.2 X (3.817)' X (.07161)^ ^
88.19 X (2.716)^ X (1.558)*
Solve, by logarithms, the following equations :
57. a; = 19V2l. 61. 38a^ = 235.
oa 62. 3V^ = 17.
58. x = -^'
Vll 63. 11^ = 13.
59. a;ViO = 95. 64. a;V5 = -5^'50.
60. a^ = 3.47. 65. 97ic2 = V855.
LOGARITHMS
151
66. Vl7a; =
67. a/5.5 a^:
: 9.74.
= V79.
'
72.
38^/^ a;V55
97 79
68. 30xV7:
= 6iA/m
73.
^^/ = V190.
69. 13A/15a
: = 27V:
Oil.
70. 26Vx =
a;V33.7.
74.
17 a;"^ = 65.
71. ^'''-.
VSx
^4l
So;
75.
xVSf ^38
75 V^
76. 5^ = 30.
81
. 65' = 3.
86. 25*-i = ll*.
77. 6'' = 75.
82,
. 40* = 5.
87. 552-- = 21*+^
78. 7^ = 15.
83,
. 18.6' = 1.86.
88. (3 + .05)^^=100.
79. 8^ = 100.
84,
. 9*+i = 15.
1
89. .9'"^ = 4.7-i
80. 4.5* = 50.
85.
. 3^-1 = 36.
90. (1.025)»«=1.01.
Calculate :
91. Iog4 20.
95. Iogi6 60.
99. log2jj.68.
92. logy 500
96. logijS.
100. logi2.3 .0423.
93. logs 35.
97. loggl.O.
101. log.5.63.
94. logi3 29.
98. Iog2o.4.
102. log.8.07.
103. Find the amount of $ 575 in 10 years at 5%, compound
interest.
104. Find the amount of $ 8500 in 12 years at 4%, compound
interest.
105. Find the amount of $3500 in 6 years at 6%, compound
interest.
106. At what rate will* $ 12,000 amount to f 14,587 in 4 years ?
152 LOGARITHMS
107. At what rate will $ 8250 amount to $ 11,627 in 10 years ?
108. What sum will amount to $520.75 in 6 years at 5%,
compound interest ?
109. What sum will amount to $817.30 in 8 years at 7%,
compound interest ?
110. In what time will $5000 amount to $8000 at 5%,
compound interest ?
111. In what time will $2750 amount to $5000 at 4%,
compound interest?
The following examples in this exercise are to be done with-
out the use of logarithmic tables.
112. Find the logarithm of 27 to the base 3.
113. Find the logarithm of 3 to the base 27.
Find :
114. log4 8. 118. logai- 122. log^^ 81.
115. log25l25. 119. logi8. 123. logaayV
116. log27 81. 120. logioolO. 124. log64 3V
117. logs 32. 121. log^27. 125. log^ooo .01.
Find a;, if :
126. log, 8 = 3. 132. log, 27 =-3.
127. log, 81 = 4. 133. log, 64 = - If
128. log, 125 = 3. 134. log, 7 = - i
129. log,6 = f 135. log, 2V = - 11
130. log, 27 = IJ. 136. log, 243 = - 2.5.
131. log,32 = l|. 137. log, J?^ = - .75.
LOGARITHMS 153
Find x,{f:
138. log, ic = 4. 142. log49a; = i 146. log27a; = — IJ.
139. logs a; = 3. 143. \og3QX = — ^. 147. logi x = — .5.
140. log9a; = |. 144. log64a; = — If. 148. logo a; = — 3.
141. log8aj = |. 145. logia; = 1.5. 149. logjg a; = — 1.5.
Write out as a polynomial :
150. logaV. ,^^ , 11 a;^'
157. log——'
151. logafe^ ^^y
152. log^^l 158. \og4.(x-yy.
153. log^. ^^^' log 8 «'(«'- ^')-
c^ 160. log3a(9-n2)i
154. log3aa;^.^ ^^^ log a (a + 5)>3 _ ^s).
155. log -2^- 162. log8a6«-\/5a=^6-^c-i
156. logL^!^. 163. log?^M±^.
Change to the logarithm of a single term :
164. log a 4- 3 log 6. 167. log 11 4- i log a.
165. log 7 — 3 log ». 168. 31oga-21og6.
166. logm+i^gT. 169. log 2 + log 3 - log 13.
170. 2 log 3 + 3 log a — log 5 — i log x.
171. log5 + 31ogaj4-ilog2/— 21og7 — ^loga.
172. 1 log 2 — J log a; + 2 log a — I log 2/ — log z.
173. log 7 + 1 log a 4- f log & - log 6 - ^ log c.
174. log (a + 1) + log (a — 1) — log 2 — J log m.
175. ilog(i) + g)+ilogO-g)-ilog(a; + ^)-ilog(a; -»!/).
154 LOGARITHMS
176. log 3 + 2 log 2 + i log 5 - log 7 - log 11 - f log 13.
177. 31og2 + l+|log7-ilog3-^log(a2 + 62)_iog^^
Find a;, if:
178. a^ = bc^.
182.
3a^-2 = d^
179. 3b' = cd'.
183.
a6- = cd'-\
180. 5m' = ?t2p2*.
184.
m^n''-^ = p.
181. a^-^ = b\
185.
(^d = l^^hn'-\
186. If log 365 = 2.5623, write log 3.65. Log .00365.
187. If log 7.008 = 0.8456, what is log 7008 ? Log 70.08 ?
188. If log 27.8 = 1.4440, write log 27800. Log .00278.
189. If log 536 = 2.7292, and log 537 = 2.7300, what is
log 5.365 ? What is log .05368 ?
190. If log 3.71 =0.5694, and log 37.2 = 1.5705, what is
log 37140 ? What is log .003717 ?
191. If log 709 = 2.8506, and log 7100 = 3.8513, find
log .07096. Find log 70.94.
192. If log 627 = 2.7973, and log 628 = 2.7980, find x, if
log X = 0.7975. If log X = 8.7978 - 10. If log x = 3.7976.
193. If log 3.35 = 0.5250, and log 33.6 = 1.5263, find x, if
log x = 9.5254 - 10. If log x = 7.5260-10. If log x = 4.5258.
194. If log 2.357 = 0.37236, log 235.8ii 2.37254, and log x
= 3.37243, find x. Also find x, if log x = 7.37251 - 10.
Given, log 2 = 0.3010 ; find the following logarithms :
195. log 4; log 40; log 8; log 800; log 32; log 3.2.
196. logV2; log ^2; logV^; logv'S; log Vi02.
197. log 5; log 50; log V5; log 2.5; log 12.5; log6J; log f .
LOGARITHMS 155
Given, log 2 = 0.3010 and log 3 = 0.4771 ; find the following
logarithms :
198. log 6; log 12; log 18; log 15; log 150; log 14.4.
199. log2V3; log3V2; logVSO; log^iOOS; log 45.
200. log 540; log .024; logS^; log 4^; log3|.
201. From log 16 how can one get log 2? Log 4? Log 8?
Log 5? Log 25?
202. From log 2 and log 15 how can one find log3 ?
203. From log 5 and log 14 how can one find log 7 ?
204. From log 50 and log 36 how can one find log3 ?
205. From log 14, log 15, log 16, how can one find the log-
arithms of all numbers from 1 to 10 ?
206. Show that there will be 31 figures in the 100th power
of 2. [log 2 = 0.3010.]
207. How many digits in 49^ ? [log 7 = 0.8451.]
208. Find a; if a*' = &.
209. Findajif a2-' = 2&. If2a2-^ = 2*.
210. Find x, (a) if .6^ = 3, (6) if .08^ = .9, provided it is given
that log 2 == 0.3010, log 3 = 0.4771.
211. What is the base if log .25 = - 1 ? If log ^ = 2.5 ?
212. Solve 22^-y = 32 and 3^+^ = 81.
213. Find x and y, if 4^+^^ = 128 and 25^"^ = 125.
214. If log5 = 0.6990, findxin the equations 2*' = 40 and
(2*)' = 40.
215. Show that log | + log f^ - 2 log .4 = log 3.
GENERAL REVIEW
Exercise 63
1. Solve the equation 12a^ — 17 x = 40 for x.
2. Tell by inspection the sum, product, and nature of the
roots of 3a^-lla; + 15 = 0, and of 3a^ + lla; = -8.
3. Find the sum of the series 4i, IJ, |, •••, to infinity.
4. Solve for x and y, x^ — jf = 152, and x = % -\-y.
5. Define quadratic equation, pure quadratic, symmetrical
expression, homogeneous expression, logarithm, arithmetical
progression, geometrical progression, alternation, composition,
and mean proportional. Give an illustration of each.
6. Form that quadratic whose roots shall be 1\ and — 2|-.
Also that one with — = for roots.
2
7. In the A. P. 15, 131, 12, ••., find the 55th term and the
sum of the first 20 terms.
8. Solve the equation a^ — 1000 = for its 3 roots.
9. Solve 2x^ — 3xy -{-y^ = 3, 3x^ — xy = 2, ioi x and y.
10. Which term in the series 2|, 3, 3|, •••, is 65 ?
11. Find all the values of ic in x^ = 9x.
12. When are the roots of la^ -\- mx + n = real ? When
equal ? When irrational ? When imaginary ?
13. From the letters in the word scholar, how many words
can be formed, of 4 letters each ? Of 7 letters each ?
166
Gi:Ni:iiAL review 157
1 _3
14. Solve 7 X ^ = 8. Are both the values of x roots ?
x^
15. Find the values of a? and 2/ in ar^— 2a;2/— a;=3, 3ic— 42/=7.
16. Find the values of x, by inspection, in
2aa;(ar'-4)(3a^ + 5) = 0.
17. What is the logarithm of 216 to the base 36? Of 8 to
the base 128 ? What is \og„ ^J^ ?
18. It a:b = c: d, prove that
19. Write the equation 32^ ^ = 64 in logarithmic form. What
is the number? The logarithm? The base? The charac-
teristic? The mantissa? Write logs; 243 = 1| as an equation.
20. How many parties of 7 each can be selected from a
school of 25 girls ?
21. Solve 2-\/x + 3 — Va; — 2 = V3 x — 2. Are both values
of X roots of this equation ? Why ?
22. Find the 6th term and the sum of 7 terms in the series
-4,6,-9,....
23. Solve 1^=^-2 -^^ = 0.
S-x x+2
24. Prove the formula for sum of a G. P. if the first term,
last term, and ratio are given. Derive formula for sum of
infinite geometrical series.
25. Solve a (6 a — 13) — 2 ax (n — x) = 5(x-\- 3).
26. Why cannot a negative number have any logarithm pro-
vided the base is always positive ? How are operations with
negative numbers performed by aid of logarithms ?
27. Expand (2-Vx - x^^/yy.
158 GENERAL REVIEW
28. Find the values of a that will make the equation
4:X' — lox—ax-^a-{-20 = have equal roots. Prove your
values correct.
29. In an A. P., Z = 14, w = 40, s = 430 ; find a and d
30. Solve 2Va;-2 + V2a; + 3 = V8a; + l.
31. From the letters in the word sweetest, how many arrange-
ments can be made, taking all the letters every time ?
32. Find the middle term of (a^y — ^Vx~'^y^.
33. Solve a^ -{- xy — y^ = 1, xy-\-2y^ = 3.
34. Insert 5 geometrical means between 2| and 30f.
35. Prove the formulas for / in the progressions. Also
prove the formulas for sum in arithmetical progression.
36. Solve for x, 2 V3a;-2 - 3 ^3x^-2 = 2.
37. Compute by logarithms, Y^^'^ ^ ^-^^ ^ -
</.0716 X 438.6
38. Form the quadratic whose roots will be —a and ^-^ —
The quadratic whose roots are -^ — ^^— —
39. Distinguish between mean proportional and third pro-
portional. Find the mean and the third proportional to a
and 2ab.
40. From a class of 8 boys and 10 girls, how many groups
of 2 boys and 3 girls each can be selected ?
41. Solve 2^^ -3 = -^ + ^.
3x + 4: 6a;H-3
42. Insert 23 arithmetical means between 3 and 63. Also
35 means between ^ and 27.5.
43. Solve x-\-2y = 2, a^ + Sf =26.
GENERAL REVIEW 159
44. Solve (x^ + x-5y-S(x^-\-x) +4:7 = 0.
45. From 16 consonants and 4 vowels, how many words can
be formed, each consisting of 3 consonants and 2 vowels ?
46. Compute by logarithms the value of J -8034 x V TO gSTg ^
.5138 X ^.00175
47. A man agreed to dig a well at the rate of 25 cents for
the first yard, 50 cents for the second, 7.5 cents for the third,
and so on. Upon completion he received $30. How deep
was the well ?
48. Find the value, by logarithms, of 2^ x (i)^ X^/^X VJ.
49. What is the value of (x - V2y + (a; + V2)* ?
50. How many figures in Q5^ ?
51. Solve a^ — a;?/ + 22/ = 4, 2x-\-3y = 5a-}-l.
52. li a:b = c: d, prove a^ -f 4 6- : c^ + 4 c?^ = a6 : cc?.
53. Prove the formulas for the number of permutations and
the number of combinations of m things taken ?i at a time.
54. Form the quadratic whose roots are 0, — — . Also the
one whose roots will be ^ — — — —
7
55. Solve (2 a; -3)2 -(a; -1)2 = 5.
56. Express as a polynomial, log ^ a^^/b ; log J x^s/S y.
57. Compute by logarithms, — ■T\\h7;:7:'
^ -^ ° . ' 101 ^'.SOO
58. Simplify {-y/l- o^ J^lf - {■y/l-x' -If,
59. How many combinations, each containing 6 white balls
and 5 red balls, can be selected from 14 white and 10 red balls ?
60. Solve 2a^ + 3a;2/-52/2 = 4, 2a;?/ + 3/ = -3.
160 GENERAL REVIEW
61. Find x that will make 2x — l, x-^7, 3a;-f-l, in A. P.
Find X if they are in G. P.
62 . Find s that will m ake the roots of 3 sor^ + 2 sa; + 9 a; + 8 =
equal to each other.
63. li a : b = c : d, prove
Va=* H- b' : Vc' + d' = i/a' - b^ : ^/(^-(j^.
64. Solve 8a.--3 + llVaP = 54.
65. Twenty men are going to march four abreast. In how
many ways can they place themselves ?
66. Expand (V2a-Va;)®.
67. Solve y-l = x-- = -'
X y 2
68. Solve 2 a^-aa;-a2 =2 (3 a; -2), for ir.
69. Find, by logarithms, the value of x, if 7^ = 100.
70. Without solving the equations, tell what is the sum,
product, and character of their roots : (a) 3a^ — 7a; = — 2;
(6) 5ar' + 4a; + l = 0.
71. What is the value of 2.7181818 -. ?
72. Solve 2a^-5a;-V2a^-5a; + l = l.
73. Form that quadratic the sum of whose roots is 11 and
whose product is 13^. Prove your answer.
74. If a:b = b:c=c:d, prove that a-{-b:b-{-c=2 a — b:2 b—c.
75. ^o\Yeabx'-a\x-l) = b\x-\-l).
76. Find all the values of x and y in the equations (x-\-yy
-x-y = Q>, x'y'^ + lxy + 12 = 0.
77. The sum of 35 terms of an A. P. is 490 and the com-
mon difference is J. Find the first and last terms.
GENERAL REVIEW 161
78. If m and n are the roots of ic^ — 07 + 1 = 0, show that
m^-{-n^ + 1 = 0. Show that i + ^ = l. That m-n = V^^.
m n
79. The sum of five terms of a G. P. is 5|^ and the ratio is
|. Find the first and last terms.
80. Find the four roots of 8 a;^ = 27 x.
81. Prove that the roots of ax^ -\- 2 bx -{- c = will be equal
if 6 is a mean proportional between a and c. When will the
roots be rational ?
82. Expand (1 — 4 x)~^ to 5 terms.
83. Find the limit of the sum of 1 — | + ^.•. to infinity.
84. Solve4cc + 4V3a^-7a; + 3 = 3ar^-3a; + 6.
85. Find two numbers whose difference, sum, and product
are to each other as 2:3:5.
86. There are three numbers in A. P. whose sum is 3. If
3, 4, and 21 be added to them, respectively, the sums form a
G. P. Find the numbers.
87. How many terms of the series 32, 48, 72, •.., amount to
665?
88. Form the equation whose roots are double the roots of
x'-3x-\-2^ = 0.
89. The 5th term of an A. P. is — 3 and the 15th term is 17.
Find the sum of the first 20 terms.
90. How many arrangements can be made from the letters
in the word holiday, taken all together ? How many, if the
three letters lid are never separated ?
91. If the base of a system of logarithms be 4, tell the loga-
rithm of each: 16; 8; 32; 2; J; 1; i; |; ^; Vj; </l',
R. & S. EX. IN ALG. 11
162 GENERAL REVIEW
92. Solve i + i = l, 1+1 = 13.
X y 0^ y^
93. How many terms of the series 41 4, 3|, •••, amount to
21?
94. Find that G. P. the sum of an infinite number of whose
terms is 4 and the second term is \.
95. Compute the 5th root of 4.281 x V.09m
logarithms. 321.7 x •v/.008074 ^
96. Find the 10th term of z^ by the binomial
theorem. (l-2Va^)^
97. Find all values of a; in 1+2 a^+3 x=3V2 a^+3 x—\.
98. If a, 6, c, dy are in continued proportion, prove that
lo? + mV + ne : Ih'' -\-m&^ nd? = ac? + &c : 2 cd.
99. Solve this equation for the value of a;: a; + - = - + -.
a X h
100. Find a: and ?/ in a^ + / = 3^, x-^ + 2/"^ = If
101. Find the sum of all numbers between 10 and 500
exactly divisible by 7.
102. What is meant by " completing the square " ? How is
it done ? What is an imaginary ? What are conjugate im-
aginaries ? Prove that log PQ = log P + log Q.
103. From the usual formulas for I and s of an A. P. derive
a formula for a not containing s. A formula for I not contain-
ing a. A formula for d not containing I.
104. Out of 15 consonants and 5 vowels how many words
can be made each consisting of 4 consonants and 3 vowels ?
105. Find, to 3 decimal places, the logarithm of 65 in a
system whose base is 15.
GENERAL REVIEW 163
106. Solve Vx + y = Vy-\-2, x — y = 7.
107. When are 3 quantities in continued proportion ?
Prove that if a, h, c, are in continued proportion, then a, a + b^
a + 2 6 + c, are also in continued proportion.
108. Find a; if (a^ + 2 xf -lSx(x + 2) +45 = 0.
109. If the base of a system of logarithms is a, what is log a?
log-? loga^? logVa? log A/a*? logAZ-i?
X y ^ a^ y^
110. Solve ±-i: = 2, -,--, = 3i
111. Calculate by logarithms the mean proportional between
V5:082 and a/.009116.
112. In a G. P. the 5th term is 12 and the 11th term is 768.
Find the 3d term and sum of 9 terms.
113. Solve =l-i + i.
a-^b — x a X b
114. Find a; if 2.5' = 75.
115. How many numbers, of 6 different digits each, can be
written from our 9 significant digits ?
116. Find x and y if a^ — a;?/ = 1^- and xy-{-y^ = 1.
117. Insert 5 geometrical means between 5 and 3645. Also
69 arithmetical means between 5 and 3645.
, , o n ^ ^1, 1 f / - 4.116 X 75.38 X .0567 \^
118. Compute the value of ( — — , ^ ^^^. — 7:7^1^ '
^ \S1.24: X ( - 1.909) X .0053;
11-9. Solve for x and y : x^-{-xy + 2/^ = 13 ; a^ + a^/ + 2/^ = 91.
120. Find x and y in x^ + y^-{-x-\-y = 26; ccy = — 10.
,«-. ai J? a.' — 1 -, 3a!-|-2 ^
121. Solve for x : — 1 -!— - = 0.
a; — 3 a; + 3
164 GENERAL REVIEW
122. Given log 40 = 1.60206, find log 2; log 5; log 20; log
50; log V5; log ■y/2.5', log i; log ^ ; log |; log 1.25; log 2i.
123. li S'.t = u'.v = w:x = y:Zf prove that
s-\-u-\-w-\-y'.t-\-v-\-x-\-z = s:t — etc.
124. Find all* the values of a; in : 3 a^x (x + 1) (a;^ - 81) = 0.
125. Solve for x and y: x^ -\- y'^ -\- x — y = 2Q -^ xy -\-16 = 0.
126. From a delegation of 15 Protestants and 11 Catholics,
there is to be chosen a committee of 6 Protestants and 4 Catho-
lics. In how many ways might this be accomplished ?
127. lix^ -\-y^ = ^ and ic + ?/ = 41, find x and y.
128. If V3a;-2 + VaJ + 2 -4 = 0, find x and discuss its
values as roots.
129. Given a = 2, Z = ^, s = \2^, find r and w.
130. Expand (2 - V^^)^
131. Solve cc^ — 2/^ = 7, a; — 2/ = 133.
132. The 5th term of a G. P. is 336 and the 9th term is
5376. What is the 2d term ?
133. Finda::^±^^^^ = :^^^+1. Discuss its values.
X — V12 a — x -y/a — l
134. If a certain number is divided by 8, the result will be
the same as if 16 were divided by the number and 3J added
to the quotient. What is the number ?
135. If V :x = x:y = y :z, prove that x-^-y is a mean pro-
portional between v-\-x and y -\-z.
136. If aj + 2/4-l = a^-/ + a:2/-l = 0,finda;andy.
GENERAL BEVIEW 165
137. What are the values of x in the equation
^ — ~ =4? Are these values both roots?
138. Solve x^ — 2ax — 2bx + (a + b-\-c)(a-\-b — c) = 0.
139. The sum of the first two terms of a G. P. is 72 and the
sum of the next two is 8. What is the 1st term ? The 5th
term ?
140. Expand to 4 terms : (y~^ — 5 a?^ —1)'^.
141. Find the series in which d = 8, 1 = 147, and s = 1425.
142. If the ratios I : m, n : p, q : r, are equal, prove that each
is equal to J ''+/+/,
m^ -{- p'^ -{- 7^
143. Solve (3 a; - 2)2 4- (« - 1) = 84.
144. Prove the binomial formula for positive integral
exponents.
145. What is the 7th term of (x-^ - ^Vxf)'^ ?
146. ^olYex^-\-xy-^2y^ = ll = a^-\-§y\
147. Find a; if 2^' = 500.
148. From the figures 1, 2, 3, 4, 5, 6, 7, how many numbers
can be formed of 5 different figures each ? How many of these
will contain a 3 ? How many will have 3 and 6 together ?
How many will be odd ?
149. Multiply .03716^ by 1.8716^ by logarithms.
150. If, in an A. P., a = s = — | and n = 20, find d and L
151. Find that G. P. whose sum to infinity is 1^ and whose
2d term is ^.
152. Solve -^ fl + J_^a; + i + l = 0.
m -{- n \ mnj m n
166 GENERAL REVIEW
153. Solve - + i = -^, a;v = 54.
ic 2/ 18
154. Insert 6 geometrical means between 5 and —640.
155. Find x and y if 3* 5^ = 75, 2* 7^ = 98.
156. How many different sounds can be made by striking
16 keys of a piano, 3 at a time ?
157. If 2(V^-3)2-3 = V^, find x,
158. Expand (2V3+3V2)^
159. Find x and y from x + y = ^, x^y^ — 2^xy = — 192.
160. Solve 6a^-3a; = 2 + V2a^-a;.
161. There are two fractions the sum of whose denomina-
tors is 5. The numerator of the first is the square of the de-
nominator of the second, and the numerator of the second is
the square of the denominator of the first. The sum of the
fractions is 5|. Find them.
162. The sum of an infinite number of terms of a certain
G. P. is 4i The sum of the first two terms is 2|. Find
the series.
163. If log 13 = 1.1139, what is log a/1300 ? log V.0013?
164. Find a;: Va^-fl: \/2x = ^ ^
x a;-f 4
165. Evaluate the decimal 1.4363636—.
166. Solve 9a;-3x2-4Vaj2-3a;-f 5 = 0.
167. Out of 7 consonants and 4 vowels how many words
can be formed, each consisting of 3 consonants and 2 vowels ?
168. Solve for x : (x-^ + i)"' = 27 and (a?"? ^ |)-i = - \.
GENERAL BEVIEW 167
169. Find the numerical value of
i logs 9-2 log27 3 + log, a? - log, 1.
170. Show that ma-{-nb: pa-\-qb=7nG+nd: pc-{-qd provided
a, b, c, d, are proportional.
171. Find a; if 4*^-1 = (!)*-«. [Without tables.]
172. How many games must be played in a league of 10
baseball clubs, provided each club plays 10 games with every
other club ?
173. Insert 5 arithmetical means between — 7 and 77.
174. Find the values of x, correct to two decimal places, in
the equation ic^ — 2 a; — 2 = 0.
175. Solve iJc^ + 5xy + Sf = S, Sx^-^7 xy-{-4:y^ = 5.
176. If the series 12, 9, •••, is arithmetical, find the sum of
20 terms. If geometrical, find the sum of an infinite number
of terms.
177. Solve a;^ + a;^ + 1 = 0.
178. Tell sum, product, and nature of roots of 3 a^ — 4 a; — 11
= 0. Also of 2a^ + 3ajH-7 = 0.
179. Find the sum of all the odd numbers between 20 and
220.
180. Expand (J V3 + W^-^Y and simplify.
181. If a, b, c, d, are proportional, show that ab + cd is a
mean proportional between a^ + c^ and b^ + d\
182. Solve the equations — 1--= — , a^+y^=a\
X y xy
183. Find two numbers in ratio of 7:5, the difference
between whose squares is 96.
168 GENERAL. BEVIEW
3/ =
184. Find the 8th term of the expansion of va — 6Vaa;, by
the binomial theorem.
185. Solvefor?/: l+V8^-32/2 = 22/.
186. Solve a; 4- 2/ + 2 Va; + 2/ = 24, « - ?/ + 3 Va; — 2/ = 10.
187. Find the value oiz: o?z-2h^= ah
188. Solve? + - = 5, ^, + -^-^ = -19.
X y x- xy y^
Hint: Let- = w, - = w.1
L X y A
189. Discuss the values of x in the equation,
V3a; + 1 = V9aj + 4 - V2 a; - 1.
190. Find, by logarithms, the values of x and y^ if 3"= = 50 ^
and 2^ = b y.
191. Find the (r + l)th term of (1 - x)^.
192. Simplify (2 - Vl - xy + (2 + Vl - x)\
193. Compute, by logarithms, the value of
^moiW^-
194. Solve Va — a; + V— (a^ + ax) = -^=
X
195. If a, 6, and c are in G. P., show that , -, and
aremA.P. « + *2' 6 + <=
196. Find the 3 cube roots of unity ; i.e. solve the equa-
tion x^ = 1.
GENERAL BEVIEW 169
197. Solve for x:-\ h - ^ = ^^, — 7^ —
2 I 1 — a x\ x(a- — 1)
198. Solve V5a; + l + 2Vaj-2 = 3V2(a;-l).
199. Solve (m + xy — (m — xy = x.
200. Of how many terms does an A. P. consist in which
d = 'S, 1 = 302, a = 5?
201. The sum of an infinite number of terms of a G-. P. is 15
and the second term is 3^. Find the fourth term.
202. Solve ^-^ = ^.
a^ - 1 x' + l 20
203. Expand (V2«— ^3»)*.
204. The difference between two numbers is 32 and the
arithmetical mean exceeds the geometrical mean by 4. Find
the numbers.
205. By the principles of proportion find x if
Va; + 2 - Va; - 3 ^ Vr) .c + 1 - 4 V.t - 6
■y/x + 2 + ^x^^ V5a.- + l + 4Va;-6
206. Find sum of i — 1 — f ••• to 29 terms.
207. Find sum of 1 + J + !••• to 6 terms.
208. Solve aV — oi^ — d^x — 5ax = 6a^.
209. The sum of 10 terms of an A. P. is 100 and the sixth
term is 11. Find the second term.
210. Solve a^_a; + l + -_-l =^.
ar — a^ + 1 3
170 GENERAL REVIEW
211. The sides of a right triangle are in A. P. Prove that
they are proportional to 3, 4, 5.
212. Find a; if
213. How many terms of the series IJ, 1, |, •••, must be
taken, that the sum may be zero ?
214. Solve 2a^-3x + 4 = -— -4 ^•
2x'-Sx + 2
215. In an A. P. a = 7, d = — i s = 55. Find n and Z.
216. If V<x + a; : Va + Va + x = Va — a; : Va — Va — x,
find a;.
Solve the following simultaneous quadratic equations and
associate the corresponding values of x and y:
217. y^ + xy-^y = -6', x'-\-xy-^x = S,
218. x(x -h 2/) = 10 ; 2/(2/ - x) = 3.
219. VaH^— Va; — 2/ = 2; 3aj-22/ = 7.
220. 3,-3-1 = 316; a;-i=4.
r y
221. a^ + 2 a^ = - 21 ; y^ ■^2xy=-b.
222. «2 + 2/2_4aVa^ 4- 2/^ = 5^2; y? - f :=! a\
223. a(a; — a) = 6(2/ — 6) ; ax -{- by = xy.
224. 6a^y^ + 5xy = 6; x -\- SO y = 12.
o«, i (^-32/)^-8a; + 24^ = -12;
''^^' ^2(2a; + 2/)2-22a;-lli/ = ^5.
GENERAL REVIEW 171
226. x-y-^x — y = 2; a?-f = 20^.
227. 2:f?-xy = \2', a? -2 xy + Zy'' = ^.
228. ^x^-{-6xy — 3x — y + 2 = 0'^ 2x-\-5y= — 4:.
229. One root of the equation a^ — 6ic+29=0is 3— 2V^^.
Without solving this quadratic, find the other root and prove
your answer correct.
230. Solve: ^- ^=2£l^.
a;— 1 aj-f-l a^ — 2a
231. Find the limit of the sum of the series 5 — 3 + 1 — ^
+ ^ — • • • to infinity.
232. Solve: V3a^-2a; + 4 = 3 0.-2-23;- 8.
233. If w, X, y, and z are proportional, show that -\/w^ — y^y
Vor — z^, w-\- y, X + z, are also proportional.
234. Find the ninth term of (ia^y-2 ^/xy\
235. Divide 9 J into three parts in G. P. such that the sum
of the first two is to the sum of the last two as 3 to 2.
236. For what values of m will the equation 4 ic^ — 15 ic + 36
= m(x — 1) have equal roots ? Verify your results.
172
LOGARITHMS
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0000
0414
0792
1139
1461
0043
0453
0828
1173
1492
0086
0492
0864
1206
1523
0128
0531
0899
1239
1553
0170
0569
0934
1271
1584
0212
0607
0969
1303
1614
0263
0645
1004
1336
1644
0294
0682
1038
1367
1673
0334
0719
1072
1399
1703
0374
0766
1106
1430
1732
15
16
17
18
19
1761
2041
2304
2553
2788
1790
2068
2330
2577
2810
1818
2095
2355
2601
2833
1847
2122
2380
2625
2856
1875
2148
2405
2648
2878
1903
2175
2430
2672
2900
1931
2201
2465
2696
2923
1969
2227
2480
2718
2945
1987
2263
2604
2742
2967
2014
2279
2629
2766
2989
20
21
22
23
24
3010
3222
3424
3617
3802
3032
3243
3444
3636
3820
3054
3263
3464
8655
3838
3075
3284
3483
3674
3856
3096
3304
3502
3692
3874
3118
3324
3522
3711
3892
3139
3345
3541
3729
3909
3160
3365
3560
3747
3927
3181
3385
3579
3766
3945
3201
3404
3698
3784
3962
25
26
27
28
29
3979
4150
4314
4472
4624
3997
4166
4330
4487
4639
4014
4183
4346
4502
4654
4031
4200
4362
4518
4669
4048
4216
4378
4533
4683
4065
4232
4393
4648
4698
4082
4249
4409
4664
4713
4099
4265
4425
4579
4728
4116
4281
4440
4594
4742
4133
4298
4456
4609
4767
30
31
32
33
34
4771
4914
5051
5185
5315
4786
4928
5065
5198
5328
4800
4942
5079
5211
5340
4814
4955
5092
5224
5353
4829
4969
6105
5237
5366
4843
4983
5119
5250
5378
4857
4997
6132
5263
6391
4871
5011
6145
5276
6403
4886
6024
6169
6289
6416
4900
5038
5172
5302
6428
35
36
37
38
39
5441
5563
5682
5798
5911
5453
5575
5694
5809
5922
5465
5587
5705
5821
5933
5478
5599
5717
5832
5944
5490
5611
5729
5843
5955
5502
5623
6740
6865
6966
5614
5635
6752
5866
6977
5527
5647
6763
5877
6988
6539
5658
6776
6888
6999
6551
5670
6786
6899
6010
40
41
42
43
44
6021
6128
6232
6335
6435
6031
6138
6243
6345
6444
6042
6149
6253
6355
6454
6053
6160
6263
6365
6464
6064
6170
6274
6375
6474
6076
6180
6284
6385
6484
6086
6191
6294
6395
6493
6096
6201
6304
6406
6503
6107
6212
6314
6415
6513
6117
6222
6325
6425
6522
45
46
47
48
49
6532
6628
6721
6812
6902
6542
6637
6730
6821
6911
6551
6646
6739
6830
6920
6561
6656
6749
6839
6928
6571
6665
6758
6848
6937
6580
6676
6767
6857
6946
(5590
6684
6776
6866
6955
6599
6693
6785
6876
6964
6609
6702
6794
6884
6972
6618
6712
6803
6893
6981
50
51
52
53
54
6990
7076
7160
7243
7324
6998
7084
7168
7251
7332
7007
7093
7177
7259
7340
7016
7101
7185
7267
7348
7024
7110
7193
7275
7366
7033
7118
7202
7284
7364
7042
7126
7210
7292
7372
7050
7135
7218
7300
7380
7059
7143
7226
7308
7388
7067
7152
7235
7316
7396
LOGARITHMS
173
No.
1
2
3
4
5
6
7
8
9
55
56
67
58
59
7404
7482
7559
7634
7709
7412
7490
7566
7642
7716
7419
7497
7574
7649
7723
7427
7505
7582
7657
7731
7435
7513
7589
7664
7738
7443
7520
7597
7672
7745
7451
7528
7604
7679
7752
7459
7536
7612
7686
7760
7466
7543
7619
7694
7767
7474
7551
7627
7701
7774
60
61
62
63
64
7782
7853
7924
7993
8062
7789
7860
7931
8000
8069
7796
7868
7938
8007
8075
7803
7875
7945
8014
8082
7810
7882
7952
8021
8089
7818
7889
7959
8028
8096
7825
7896
7966
8035
8102
7832
7903
7973
8041
8109
7839
7910
7980
8048
8116
7846
7917
7987
8055
8122
65
66
67
68
69
8129
8195
8261
8325
8388
8136
8202
82(57
8331
8395
8142
8209
8274
8338
8401
8149
8215
8280
8344
»407
8156
8222
8287
8351
8414
8162
8228
8293
8357
8420
8109
8235
8299
8363
8426
8176
8241
8306
8370
8432
8182
8248
8312
8376
8439
8189
8254
8319
8382
8445
70
71
72
73
74
8451
8513
8573
8633
8692
8457
8519
8579
8639
8698
8463
8525
8585
8645
8704
8470
8531
8591
8651
8710
8476
8537
8597
8657
8716
8482
8543
8603
8663
8722
8488
8549
8609
8669
8727
8494
8555
8615
8675
8733
8500
8561
8621
8681
8739
8506
8567
8627
8686
8745
75
76
77
78
79
8751
8808
8865
8921
8976
8756
8814
8871
8927
8982
8762
8820
8876
8932
8987
8768
8825
8882
8938
8993
8774
8831
8887
8943
8998
8779
8837
8893
8949
9004
8785
8842
8899
8954
9009
8791
8848
8904
8960
9015
8797
8854
8910
8965
9020
8802
8859
8915
8971
9025
80
81
82
83
84
9031
9085
9138
9191
9243
9036
9090
9143
9196
9248
9042
9096
9149
9201
9253
9047
9101
9154
9206
9258
9053
9106
9159
9212
9263
9058
9112
9165
9217
9269
9063
9117
9170
9222
9274
9069
9122
9175
9227
9279
9074
9128
9180
9232
9284
9079
9133
9186
9238
9289
85
86
87
88
89
9294
9345
9395
9445
9494
9299
9350
9100
9450
9499
9304
9355
9405
9455
9504
9309
9360
9410
9460
9509
9315
9365
9415
9405
9513
9320
9370
9420
9469
9518
9325
9375
9425
9474
9523
9330
9380
9430
9479
9528
9335
9385
9435
9484
9533
9340
9390
9440
9489
9538
90
91
92
93
94
9542
9590
9638
9685
9731
9547
9595
9643
9689
9736
9552
9600
9647
9694
9741
9557
9605
9652
9699
9745
9562
9609
9657
9703
9750
9566
9614
9661
9708
9754
9571
9619
9666
9713
9759
9576
9624
9671
9717
9763
9581
9628
9675
9722
9768
9586
9633
9680
9727
9773
95
96
97
98
99
9777
9823
9868
9912
9956
9782
9827
9872
9917
9961
9786
9832
9877
9921
9965
9791
983(3
9881
9926
9969
9795
9841
9886
9930
9974
9800
9845
9890
9934
9978
9805
9850
9894
9939
9983
9809
9854
9899
9943
9987
9814
9859
9903
9948
9991
9818
9863
9908
9952
9996
Phillips and Fisher's Geometry
By ANDREW W. PHILLIPS, Ph.D.
and IRVING FISHER, Ph.D.
Yale University
PHILLIPS AND FISHER'S ELEMENTS OF PLANE AND SOLID
GEOMETRY $1.75
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But the feature which more than any other distinguishes it from
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By this method of illustration the great problem of educating the
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TREATMENT BY TYPE FORMS
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are presented by means of a typical example which is described in unusual
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Only after the detailed discussion of a t5rpe form has given the pupil
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