/I IN MEMORIAM FLORIAN CAJORl c^^ Oc.i<- ■ — '^ ^ }(?} '>^ Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/exercisesinalgebOOrobbrich 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 + 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 = 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„. 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{^ 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^ = 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. ( \/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^ 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. (■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^^'^ ^ ^-^^ ^ - , 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;