$B ST t,li3 THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA EDUC-PSYOI GIFT OF Emily Turner '" ANSWERS Wentworth's Algebra, vi?Mm»iL*i tTJ^JKIWiVi -^ EDUC- PSYCH. LIBRARY (pA/Sd h/W ^^Si^ d^^^^^p" ANSWERS. t^C. B^ • Exercise I. y(lf 1. 22. 6. 94. 11. 6. la 25. 21. 39|. 2. 26. 7. 81. 12. 1. 17. 64f. 22. 20t 3. 564. 8. 16. 13. 120. 18. 31. 23. 20|. 4. 6. 9. 2. 14. 25. 19. -15. 24. 0. 5. 39. 10. 127. 15. 43. 20. 24. 25. lOJ. 26. a + h. 27. 2x. 28. a -5. 29. x^\. 30. x-2, x-\ X, x-\-\ x-\-1. 32. y. 31. ax. 33. a -5. 34. ah-^c. 35. ^~^^=a5m- c -6.+ I 36. ^^ +6b(c + d-a). 37. m — a 4a^-5a2^,2_^35*. Exercise II. 3. 30, 5. 5. 60 ft. broken off; 15 ft. standing. 4. 25 cents. 6. 27, 36. 7. Horse, $168; cow, $42; sheep, 1 6. 8. Apples, 13 cents ; pears, 26 cents ; oranges, 52 cents. 9. Harness, $50; wagon, $100; horse, $200. 10. Thomas, $1.20; Richard, $1.20; Henry, 60 cents. 11. A, $100; B, $400; C, $500. Exercise III. 1. 5. 3. -11. 5. 51. 7. $2484. 40. 4. -3. 6. 495. 8. B.C. 241. D. 14. 10. 150 steps in all ; 76 steps forward. 077 ALGEBRA. Exercise IV. 1. 0. 4. 3a;2. 7. 25 my. 10. 14 aV. 2. 6mx. 5. 7my2. 8. -22aJ2. 11. Qb^m^ 3. — 20ww^. 6. 2 ah. 9. — 55^y. 12. 9 a — 106. 13. —a^c-\-xi/z. 14. 3a;^y — 4a6 — 2mn. Exercise V. 1. 9a + 9h + 9c. 3. 2a + 2^». 5. -2a+2^>+2d 2. a-b. 4, a + b + c. 6. 2a^-9ar^~8x+10. 7. ux^+4:a^-i-Sx^^2x-9. 9, 6ab-{-7ax'-9a^xi-aa^. IL -2:ry-23/2+ll2/z + 52?-6y-62;. 12. 6m' — m^n -}- 2mV + 3 mV — 4 mw* + 5 w*. Exercise VI. 1. 9. 2. -25. 3. -89. 4. 200. 5. 506 years. 6. 35° in latitude ; 75° in longitude. Exercise VII. 1. 9x. 6. Alaa^. 11. 6a^. 2. -8 ah. 7. 8 A. 12. 13 a;^^. 3. -1ab\ 8. a;y. 13. -6a^. 4. 22?nV. 9. 8a:p + 3ay. 14. -3a5 + 6ma7. 5. -4ay. 10. 2ab^y-aby. 15. 3a — 25 + 4c. Exercise VIII. 1. 4« + 2c. 2. a + 65 + 4c. 3. 2x^-2x-^. 4. 3a;*-a;3-14a; + 18. 6. -5:ry-5^z+y2+2y2 5. T?-ax-^2a^. 7. 2a«-6a25 + 6a52_253. 8. —\xy-\-2xz — 'i^-\-hyz — '^. 9. a^+7a5:r-5a:2 4.j2^_|.125^ + :rl 10. -2r^;« + 3a;y2_2/3_i4^_|_2^3/_10y2-|-6. 11, -a^-8a5^ + 5^ 12, - 2a;^ + 4a;''y-8a7?/^-8/, ANSWERS. 13. a'-W -Za%c -ZaWc- oh(? - a^(? -W(?. 14. 2a + 46-9c + (^. 15. h-a\ a^ ^ a^h -\- ^ ab^ -\- ¥ . 16. 3?-^xy-\-'l^. la -10a2-4a5 + c2. 17. 19ac+17cc?-17. 19. xy -"Ix — ^y -\-\, 20. -2a26c-2a52tf + 2a5r-2a5c. 21. 52^^-50; + 5. 22. ^ + a:2/ + 3/2. 24. 2hx-\-2hy. 23. a2(?--hf-cc(?-\-dy\ 25. 1x-ly + 2f. 26. a262+12a5c-9aa;2_4^j2_|_g^^^_3^2^_ 27. -a2-652_|.^ 29. rr^-Sy^-Szl 28. 2a2. 30. 2a3_2a2c + 2ac2. Exercise IX. 1^ 25. 7. a + a^ 13. -5 + 10c. 2 a-{-h-2c. 8. 2a + 3t?. 14. — 5 a. 3. Zx-^y-z. 9. 3a-35. 15. 4a-166-2c 4. a~h-\- c. 10. 2a- h. 16. 3a-2c. 5. -2x. 11. 5 a. 17. 16 -12:r. 6. -96-f 14c. 12. 4a; — y — z. 18. 4.C. 19. 3 a. Exercise X. 20. a. 1. (2a-35)-(4c-c?) + (3e-2/); (2a-36-4c) + (c?+3c-2/). 2. (a-2a;) + (4y-32;)-(25-c); (a-22; + 4y)-(3z + 26-c?). 3. (a^ + 3a'')-(2a3 + 4a2)+(a-l); (a« + 3a*-2a3)-(4a2-a+l). 4. -(3a + 26) + (2c-5c?)-(e + 2/); -(3a + 25-2-2c)a:-(6a + 3c)y + (4^)z. 9. (a-Q:r + (2a + 3)2/ + (4a-35-2)0. 10. la-U-2c)x-{2b-^2>c-\-a)y^{a^U-\-bc)z. 11. (12a-15c):r + (12a + 45 + 6c)y-(125-3c)z. 12. l2a-2h)x-{Zh-\-c)y. 1. - 136. 2. - 320. 3. - 18.0621. 4. 90. 9. 10. 11. 12. 13. 14. 11. -12al 45 m^n. — \2ahxy. — 8 cdmn. Exercise XI. 5. -258. 6. 45,700. 7. 61,225. 8. 82.477956. Exercise XII. 15. -10a'"+". 16. 21 aV. 17. 22^abc. 18. 19. 20. -96a2^V. - 18,954 a^^m/. -md^hSJ. Exercise XIII. 9. - 728. 10. 2052. 11. 183.92. 12. -1718.382762. 13. -8.0001703416. 21. 22. 23. 24. 25. 26. - 42 rn^qo^. - 144/^«. 24a«mV. 162a;VV. - 24 ^2^,2^, V. - 168 a*5«mW \2(^h-^ah^. 7. 6;rV-8a:2^ + 10ar^yz^ 24a*-27a3^. 8. aVy- 5 A^ -faVy + 2aa;V 9. 27a«6^ _ 9 a^js + 12 aW + 3 ah^. 10. -15a;»y + 10^y + 35:ry-5a;2^^ 12^:32^ -15a7y- 24 r'y. 12. ^a^-2a^b-a%\ 13. Sa^^y^H^^V-lSa^^y^z^+lSa^y^-Oa^yzl ANSWERS. Exercise XIV. 1. x*-[-s?~20. 4. a?-f. 7. x^ — 2x-\-l. 2. 3/^+7y-78. b. 2x'-{-2>x^j-2f. %.a?-<^a^x. 3. a^-x\ 6. 6 ^'^-96. 9. - 10 5^ - aZ)2 -f 26 a^^ - Ya^. 10. 2o^ + bah + 2h\ 13. 6a35-23a252 + 20a53. 11. a«-Z>^ 14. --^a'-^aW+l^aW-Qab\ 12. a^ + ^^ ' 15. a^ + a^h^ + h\ 16. a« - 5 a*^ + 10 a^h^ - 10 aW + 5 a5* ~ 5^ 17. :r2 - 43/2 +12y2;- 9^1 18. ^x^ + x^y~l^xif-\-2a^yz-]-2,xyh-\-^fz. 19. :r^ 4- 0!^y + n^y + ^2; + ^3/z + ^y^^ + ^^^^ + ^y^^ + 2/^2;^- 20. a2-3a5c + ^^ + ^^ 23. ar'-41a;-120. 21. a;2 + 3:ry + 3/3-1. 24. :rH 10^7-33. 22. ^+151a:-264. 25. x'-lx^^21x'-llx''-2b3?-^^a?-{-2x. 26. ^o^~bo(^-\-^x^-l03?-^x'-bx-^. 27. 25 a^Z) - 9 a'¥ + 22 a^b'' - 4 a^^^ + aV. 28. 4 a^y + 8 aV - ^4 aV - 128 a^f. 29. 6 7)1^71 — IS m^n^ -[-18 7?i3n^ — 6 mn^. 30. 24a«^>-26a^^^^+4a*^)«-6aW-5a258-f4a6»-35i». 31. a;*-10^ + 9. 32. a^ + x^+1. 33. a^ + a*b* + b\ 34. 32 a^5 - 8 a^b'^ -f 8 a^^^ - 6 a*Z>* - 2 a^b^ - a%^ + a6^ 35. a:^ + ax"^ - 27aV _ 13 ^3^ _^ 134 ^4^ ^ 120 a''. 36. 531,441 oP- - h'^ 37. - ?/ V - 3 2*. 38. ^a'b-o^b^-^aW-^a'b^+2a¥-2ab^+U*-&b\ 39. _ «4 _|_ 2c,2J2 _ J4 _|. 2 aV + 2Z»V - c^ 40. J2 _ d\ 41. 4a2+4?/-F4c2+4c?2 46. 2/r. 51. -6:r. 42. 2^2 + 252 -f 2^2. 47. 0. 52. 25a*- 25^^ 43. ac-(?. 48. 0. 53. 24a5 -12052. 44. 2mn. 49. 0. 54. ^x^y^-\-2o[?f—f. 45. 2aH25' + 2c2. 50. -3Z>a:. 55. 0. ALGEBRA. Exercise XV. 7. x'^^xy^f. 9. 4^2^4;r+l. 11. \-'lo?-\-x^. 8. /-2y2+22. 10. 4a2-f20a5+2552. 12. 9aV-24«:r«+16:r''. 13. l-14a + 49a2. 17. 144 + 120^ + 25 a;^. 14. 25a:2/+20a:y+4. 18. l^x'y'' -'^x^z^ -\-fz^. 15. a2^,2_|_2aZ,cc^+^c?2 19. 9aW- 6a5V^+5V(^2 16. 9m^n^—24:mn-j-16. 20. 16^;^- 8:ry + :r'y*. 21. ^t^-f. 24. 9 a2Z,2 _ 4 J4 27. 36 x'f - 25 y^ 22. 4:a^-b^ 25. 16:r^-92/^ 28. 16:r^«-l. 23. 9-ar^. 26. aV-^y. 29. l-9a2Z)6. 30. aV-Z>y. Exercise XVI. 1. a^ + fi-z^ + 2xy + 2xz + 2yz. 2. a;2 + 3/2_|_22_2<^y_|_2a;2-22/0. 3. m^-{~n^-{-p^-{-q^-]-2mn—2mp--2mq—2np—2nq-{-2pq. 4. a;* + 4:^ + 9 + 4a;«-6a;2_i2:r. 5. a;^ + 36:r2 + 49-12a;3 + 14a;2-84:r. 6. 4a;^ + 49:^2 + 81 -28a:3 + 36a;2_i26rr. 7. a;4_|_y4_j_24_j_2a;2^-2a:2^-2i/2z2. 8. .i;8 + 18a:y + 2/«-8^y-8^/. 9. «6_|_j6_^^6_^2a«^« + 2aV+25V. 10. ^« + 2/« + 2«-22;«2/«-2a;«2« + 22/«^. 11. x^-j-Af + 9z^ + 4:xy-&xz-12yz. 12. ,;* + 4y^ + 25 .-^ - 4 rr^y^ + lOr^.^ - 20y2^. 13. ^4^4_|_4^_8a;. 14. x' + Sdx' + ^Q-lOa^-IOx. 15. 4a;^-7.r2+16-12:r3 + 24^. 16. ^ + 42/2_|_9^2_^4^y_|_6^2; + 122/2:. Exercise XVII. 1. ^ + 5a;+6. 3. a:^- 9a;+ 18. 5.a^-1x-S. 2. ^+6a:+5. 4. :r2-9:r + 8. 6. ar* + 3 a: - 10. ANSWERS. 7. ar' + 4:x-21. 8. x^-6x + 8. 9. .T2+12r^-+ll. 10. Q!^-\-ax — 6a^. 11. x^ — (c + d)xi-cd. 12. X^—Sx7/—4:f. 13. a2_7j^4_io52. 14. x* + Sa^f-\-27/. 15. a;'^-2:r3y-3rr22/2^ 16. a^x^ — Sax — 54:. 17. rr^ + (a — Z)) :i; — a5. 18. x^ — 7x — 4:4:. 19. :r2 + ^-132. 20. :^- 15:^ + 50. o^+Sx^a+Sxa'+a^ a^~?>x'a+^xo?-a^ 0(^+^X^+^X-\-l. a^-'^x^^^x-l. Exercise XVIII. 5. x^-{'^c(?a+(Sx^a^-\-4:xa^+a\ 6. x'^-A:x^a+(ja^o?-4:xa?-\-a^. 7. rr* + 4^ + 6n^ + 4:i' + l. 8. rr^-4^3_|_g^,2_4^_|_]^_ 9. x^-{-^oxSj + lOa^f-\-lO:x^f + bxy^-\-y^. 10. .7;^ — 5a;^?/ + lOa^if — lOx^ + bxi/~ if. 11. .^'^ + 5:r^ + 10^.^+10:^2 + 5:^ + 1. 12. :r^ - 5:?;^+10:r«- 10:^2 + 5a; -1. Exercise XIX. 1. 66. 4. -214. 7. -11. 10. -.022^. 2. 581. 5. -21.7. 8. 43^. 11. .3183+ 3. -128. 6. -1.23. 9. - .1123. 12. .0101321+. Exercise XX. 5. Bm. 11. X 57/' 17. 9m y 23. 8abh\ 6. -4a^ 12. -9a^ 18. 24. - UxYz^. 7. 5a 13. -35m 4 a:?; 19. 4:mx* 5a« 25. 8b:^. 8. 1 6 a. 14. h(?. 20. Gx'z. 26. o?bx. 9. 15. mV. 21. UcV 0? 27. I1a\ 10. 7ac. 16. - 17a?/. 22. ^am^n 28. 1a^-\ vh\ 9. 4aV + 2aa; + l. 11. 16a252+ 12a5a; + 9:r2. 10. l + 3^y+9:ry. 12. 81a*+27a3+9a2+3a+i. 13. 16a* + 24a35 + 36a252_|_54^j3_^815*. Exercise XXV. 1. x^ — xy^f. 8. 64^?/2__8^^^_|.^_ 2. x^-a^y^a^f-xf+^f. 9. Sla^- 54a5 + 36^2. 3. l-2a + 4al 10. 16^2-40^5 + 10051 4. 9a2-3a5 + 52. 11. 16a252- 12a5a; + 9:r2. 5. 4aV-2a:r+l. 12. x^-1x-\-4c^. 6. a:2_3^^y_j_9y2 13^ 9a;2^-6^y2 + 4z2. 7. a^-2a35 + 4a252-8a53+165^ 14. 256 a" - 192 o?b + 144 ^252 - 108 ab^ + 81 b'. Exercise XXVI. 1. a^ + x^y-^xf + f. 6. x^-^x'y + ^xf~21f. 2. a^-a^y + xf-f. 7. 8;r« + 4^2_|_2:i;+ 1. 3. a^+a*.2;+aV+a2:f2+a:?;*+a;^ 8. ^a? -4:0^ + 2x ~l. 4. a«-a^a;+aV-a2:r3+aa;*-ar^. 9. 27aV + 9a2:r2+3aa;+l. 5. x^^2>a^y + ^xf + 21f. 10. 27aV-9a2a;2^3^^_l 10 ALGEBRA. 11. 32a* + Wa'b + Sa^b^ + 4.a'b^ + 2ab' + b'. 12. S2a' - 16a'b + 8a'b^ - 4:a'b^ + 2ab' - b'. is! a^ + Sx*i/ + 9a^i/ + 21a^f + 81x/ + 2ASf. 14. c(^ -Sx^i/ -^9a^f - 27T^f + 8lxy^ ~24S7/. 15. 27a« + 18 a^c + 12 ac^ + 8 c-^. 16. 27 a^ - 18 a^c +12a(^~8 c^ 17. 64a« + 160a2+ 400 a +1000. 18. 64a»-160a2 + 400a-1000. 19. 125a;3 + 25a;2 + 5^+l. Exercise XXVII. 1. rr^-^y^ + y*. 6. a^-x* + 1. 2. a^-a^+l. 7. 16:^;^- Axy+y\ .3. a^ — aV+a Y~ay+f. 8. 16-4 a' + a'. 4. b'-b' + b' -b'+l. 9. 81a*- 9a'b' + b\ 5. a^-a'b' + b'. 10. 81c*- 9c2+l. Exercise XXVIII. 1. X = 4:. 7. x = 6. 13. x = —7. 19. X = 4:. 2. x = 2. 8. x--=9. 14. :r = 3. 20. X = 10. 3. X=:2. 9. x=7. 15. x = 7. 21. a; = 3. 4 x = 5. 10. a; = 11. 16. x = 0. 22. X = 15. 5. x = l. 11. x = 9. 17. x = 8. 23. :r = 2. 6. x^l. 12. x = 2. 18. x = 6. 24. x = S. 25. :?; = 4. Exercise XXIX. 6. 70. 7. 43. 8. 23. 9. 21, 7. 10. 36, 26, 18, 12. 11. 8,12. 14. $68. 12. Father, 50 yrs. ; son, 30 yrs. 15. Cloth ,$3; silk, $6. 13. 30, 26, 22, 18, 14, 10. 16. 52. 17. A, $130; B, $150; C, $130; D, $90. 18. 152 men ; 76 women ; 38 children. 19. 13, 21. 20. A, 84 yrs. ; B, 26 yrs. ANSWERS. 11 21. Wife, 14000; each son, $1000; each daughter, $500. 22. 49 gallons. 24. 50, 40. 26. 24, 60. 23. $21. 25. 42, 18. 27. A, 88 years; B, 44 years. 28. 18 years. 30. A, 57 years; B, 19 years. 29. 40 years. 31. 4 years. 32. 10 dollar pieces ; 40 twenty-five-cent pieces. 33. 28 pounds of better kind ; 2 pounds of poorer kind. 34. 28 days. 35. 60 gallons @ $2 ; 40 gallons @ $3. 36. 40 children. 37. 31 J gallons. 38. $45. 39. 2 half-dollars; 6 quarter-dollars; 24 dimes; 48 half- dimes. 40. 10 half-dollars ; 41 quarter-dollars. 41. 80 guineas ; 128 half-crowns. Exercise XXX. 1. 5a(a-3). 6. ?>a%^ {^a% -la-^-^h^). 2. 6a(a2-f 3a-2). 7. 21xY{2-{-4.x'f -^xY). 3. l{lx'~^x + 2). 8. 45^y(:iV-2^-82/). 4. ^xy{x^-2>xy-\-2y"-) 9. 10ay\o?-2ay+?>f). 5. y{f-af-^by + c). 10. ^2o:'h\l + ^aW -A:a^¥) Exercise XXXI. 1. {x — a) {x — h). 6. {ab-{-pq){x — y). 2. (a — y){b-^ y). 7, (ex -f ay) idx — by). 3. {b — x)(c-\- x). 8. (ac — bd) (by — dx). 4. (m -{-a) (x-\- n). 9. (a — b)(x — y). 5. (cx + y)(dx~y). 10. (cz + y) (dz — y). Exercise XXXII. 1. (x + S)(x + 3). 5. (x + 11) (x -f 10). 2. (x + 6) (:^ -f 5). 6. (y -f 20) (y + 15). 3. (2/-M2)(2/ + 5). 7. (b + 11)(b + 6). 4. (z + 12)(z + l). a (x + 2)(x + l). 12 ALGEBRA. 9. (x + 6)(x + l). 10. (a + 8b)(a + b). 11. (x + 9a)lx + 4:a). 12. (y+16;?)(y + 3^). 13. (z + 25q){z + 4:q). 14. (a2 + 3)(a2-f 2). 16. (^ + 3)(z«+l). 16. (a5 + 16) (a^ + 2). 25, (rs + 18 26. (mV + 17. (a;y + 4)(:^;y + 3). 18. (2* + 8)(0« + 2). 19. (a + 6^»)(a + 45). 20. (a;3_|.5)(^^_|_4>^ 21. lax+llh)(ax + Sb). 22. (ac + 5a:)(a(? + 2:z:). 23. (xi/z+U)(x^z + S). 24. (^)c + 13a)(^»c + 5a). 2)(?'S + 5z). 17jO^)(77lV+3^^). Exercise XXXIII. 1. (x-5)(x~2). .2. lx-19)(x-10). 3. (a -12) (a -11). 4. (^ - 20) (5 - 10). 5. (2 -23) (2 -20). 6. {x-e)(x-l). 7. (x'-3a^)(x'-a'). 8. (a;-6)(a;-2). 9. (z- 56) (2-1). 10. (2/3 -4) (2/3 -3). 11. (^y-26)(a:y-l). 12. (a'b^-6)(a^b^~6). 13. (a5c-ll)(a^c-2). 14. (a:-10)(:r-5). 15. (:r-10)(a;-10). 16. (a^-18)(a:r-3). 17. (ax-lSb)(ax-Sb). 18. (ac-lSz)(ac — 11 z). 19. (5;-13)(:^-7). 20. (x—15y(x — 8). 21. (0-45) (0-8). 22. (x — a)(x~ c). 23. (y2; - 17ab) {yz - 11 ah). 24. (cc?- 17a5) (c?c?- 13 a^). Exercise XXXIV. 1. (a; + 7)(a:-l). 2. (:^+12)(a:-7) 3. (y + 12)(y-5) 4. (y+15)(y-3) 5. {z + 12) {z - 1) 6. (2+20)(z-7) 7. (a + 25) (a -12) 8. (a + 30) (a -5). 9. (5^ + 4)(Z>^-l). 10. {be -\-li) {be -11). 11. (c^ + 20)(c-^-5). 12. (d? + 30)(c-13). 13. (a + 12) (a -11). 14. {xyz^ll){xyz — 2). ANSWERS. 13 Exercise XXXV. 1. (x -1)(x + 4). 9. (y - 10a) (y + 5a). 2. (y - 9) (2/ + 2). 10. (ab - 4) (a^ + 1). 3. (x~12)(x + S). 11. (a:c - 9) (ao; + 6). 4. (2 - 15) (0 + 4). 12. (cc? - 30) (cd -f 6). 5. (z - 14) (2 4- 1). 13. (a^c - 2) (a^c + 1). 6. (a - 20) (a + 5). 14. (/z^ _ 12) (yV + 7). 7. (c^ - 10) (c« + 1). 15. (ab - 18) (a5 + 2). 8. (:r-10)(a: + 2). 16. (x -a) (x + b). Exercise XXXVI. 1. (x + ey. 5.(2/4-100)1 9.(^+12)2. 2. (:r + 14)2. Q (■^ ^ 7)2_ 10^ (^2 _^ gl)2^ 3. (:r + 17)'. 7. (:r + ISyf. 11. (2 a + 3 ^>2)2. 4. (z + 1)1 8. (y2 + 8 2^)2. 12. (Sx2^ + 5 z)\ 13. (3:c + 2y)2. 14. {2a^x -\~ba^y)\ Exercise XXXVII. 1. (a -4)2. 9. (^3-17)2. 17. (3mn-4)2. 2. (a - 15)2. 10. (2 x'y-b fzf. 18. (2 ^»a; - 3 xyf. 3. (:r-19)2. 11. (4:r3j2-y22>)2 j9^ (7a-8^.)2. 4. (a: -20)2. 12. (Sabc-bcdf. 20. (8a;2^- 10072^)2. 5. (y-50)2. 13. (4.:^-xfy. 21. (7a^^-2a;)2. 6. (y2 _ 10)2. 14^ (^3^ _ ^^^y 22. (11 0:2 _ 13 ^y^ 7. (y - 25z)2. 15. (6a;y - 5y2)2. 23. (17a:yz - Syzdf. 8. (x^- 16 y2)2. 16. (1-3 aby. 24. (19 a;yz - 2 a5c)2. Exercise XXXVIII. 1. (a + b)(a-b). 5. (a2 + l) («+ 1) (a- 1). 2. (a + 4) (a - 4). 6. (a^+^*) (^2+^2) (ai-b) (a-b), 3. (2 a + 5) (2 a - 5). 7. (a*+l) (a'+l) («+l) (a-1). 4. (a2H-52)(a+5)(a-5). 8. (6 ^ + 7y) (6 a; - 7y). 14 ALGEBRA. 9. (10xi/+llab)(10xy-llab). 10. (1 + 7x) (1 - 7x). 12. (a-b + c)(a-b~ c). 11. {a^-\-bb){a'-bb). 13. {x + a-b)(x-a + b). 14. (a + 5-f c + c?)(a+^ — c-c?). 15. ^xy. 17. (2r + y + z)(a:-y-z). 16. {l + a-b)(l-a-\-b). 18. (a:-y + z)(a:-y- z). 19. (a + 25 -3c) (a -25 + 3c). 20. {a~y-\-x-['z){a — y — x — z). 21. (2; + a;-y)(2;-^ + y). 22. {x — y -{- z -{- d) (x — y — z — d). 23. (a; — z + y — a) (a; — z — y + a). 24. (a + 5 + c)(a + 5 — c). 25. (a — 5 + ;r — y) (a — 5 — a; + y). 26. (a:r+5y+l)(aa7 + 5y — 1). 27. (l+^-y)(l-^+y). 30. (:^ + y+2)(a;-y). 28. 12(a-l)(2a+l). 31. {x-\-y) {x-y-\-2). 29. (a-5 + a;)(a-5-5;). 32. (d + x -2y){d-x + 2y). 33. (a + Z, + c)(a-5-c). 34. (2^ + 3a;-l)(2:r2-3a;+l). Exercise XXXIX. 1. {a-b){a'+ab + b% 3. (a;-7)(:^+7a;+49). 2. (a;-2)(a;2_|_2^^4) 4, (y- 5) (y^ + Sy + 25). 5. (y-6)(y^ + 6y + 36). * 6. (2x-2>y){4:a?-{-^xy-\-^y'). 7. (4y-10z)(16y2 + 40yz + 10022). 8. (9a;-8y)(81:^;2_^72:ry + 64y2). 9. (3a-12)(9a2 + 36a+144). 10. (10a-115)(100a2 + 110a5 + 12152). Exercise XL. 1. {x-\-y){o(^-xy-^lf). 3. (^+6)(^-62:+36). 2. (^+2)(^-2a:+4). 4. (y+4z)(y2-4yz+16z2). ANSWERS. 15 5. (U + bc) (16 b' - 20 be -j- 25c'). 6. (6a + 8c)(36a2-48ac + 64c2). 7. (9x + 12i/) (81a^-108xi/-i-14:4:y^). 8. (x + 2/) (;r'* — :r^3/ + cc^y^ — :?;?/^ -f y"*). 9. (x + y)(ix'-x'y-i-xY-a^y'i-ar'y'-xy + y'). 10. (2^ + 3c) (165* - 24:b'c + 36 Z^V- 546^3 + 81 c'). Exercise XLI. 1. {a^ + b')(a'~a'b'^ + b'). 2. i^a" + b') (a' - a«52 _|. ^4^4 _ ^2^6 ^ ^s^ 3. (x^ + y*) (a:» - xY + y^). 4. (P + 4c')(b'-Wc'-}-16c''). 5. (ri-2+l)(;r*-a:2_^^ 7^ (4a2+:r2)(16a*-4aV+:^*) 6. (a' + 1) (a« - a* + 1). 8. (9 + c') (81 - 9 ^ + c*). Exercise XLII. 1. (a^ + ab + b') (a' - ab -{-b'). 2. (Sa^ + Sxy + 2f)(Sc(^-Sxi/ + 2y'). 3. (Ax^ -\-Zxy-f) (4:r2 _ 3^,^ __ ^-^ 4. (9a2+7a^.+452)(9a2-7a5 + 452). 5. (9a2+10a5 + 4^>2)(9a2_10a5 + 4^>2). 6. (3a;2_|_2a:y+7y2)(3^-2:ry+7y^). 7. (5 a^ + 7a5 + 4 ^2) (5 a^ - 7a5 + 4 b'). 8. (7m2 + 4 mw + 9 n^) (7m2 — Amn-\-9n'). 9. (3a2-f 3ac+5^)(3a2-3ac4-5c2). 10. (7a2+13«^> + 115')(7a2_13a6+115«). 11. (8a;2_^4^^_^9^2)^8:c2-4:ry + 93/'). 12. (2:r2_|_5^^_3^)^2^-5^y-3y2). 13. (60^ + xy - 4:f) (5x^ - xy -4^). 14. (9:z;2^4^y_^>)^9^^_4^^_^^^ Exercise XLIII. 1. (4a;+l)(3rr--2). 3. (4:r+l)(3:r-l). 5. (a;+2)(3a:-2). 2. (3^-1) (4^-1). 4. (a;4-l)(3:^-5}. ^ (Sx-\-4:)(2x-l), 16 ALGEBRA. 7. (4.x + 1) (^ + 3). 16. (a + 2x){a- Sx). 8. (x + S){^x-l). 17. (2a + 55)(4a-35). 9. (2^+1) (2a; -3). 18. (3a- 2c) (2a- 5c). 10. (a;-a)(a:-2a). 19. (4a; + 3y) (2a;+ 7y). 11. (3a2 + x") (4a2 - x"). 20. (4^; + 83/) (2x - 7y). 12. (2a; + y)(a; + 2y). 21. (2.r + 7^/) (3a;-y). 1.3. (2aa: + l)(3aa;-l). 22. (Ua~h) (a- 2b). 14. (3 5 + a;) (2 5 -3a:). 23. (2c - d) (c - Qd). 15. (2a; +1) (2a; -I- 3). 24. (23/ + 3z) (3y-z). • Exercise XLIV. 1. (a + hf. 4. (x + yy. 7. (a; + y + z)2. 2. (a + iy. 5. (a;-l)^ a (x-y-zf. Z.(a-Vf. ^.(a-c)\ 9. (a + 5~c)2. Exercise XLV. 1. (2a;-y-3)(a;-2y-7). 8. (2a;+y-3z)(3a;-y+3z). 2. (6a;— y+l)(a;— 6y— 1). 9. (6a;— y-2;)(a;— y+62). 3. (3a:+2y+l)(2a;-3y-l). 10. (5a: — 3y + 22)(a; — y-z). 4. (5a;-3y+2)(a;-y+l). 11. (ar+y+z)(2a;-3y-2z). 5. (a:+y-3)(2a;-3y-2). 12. (2a:-3y+2z)(3a;-2y+3z). 6. (a:+5y— 3)(a;~5y— 7). 13. (a; — y) (a; — y + 5). 7. (2a;-y+z)(a;-2y-z). 14. (2a;-y)(a;+3y-2z). Exercise XLVI. 1. 5(a;+l)(^-4). 9. (a^ -\- a-\-Y)(o? - a-\-V). 2. 2a:«(a;-2)(a;-6). 10. (x -y)(x^y~'z\ 3. 3(a5 + l)(«^-4). 11. (a-h)(h~e). 4. (a + a:)(a + a; + 4). 12. (3 a; - y) (a; - z). 5. (a — h-\-c)(a — h — c). 13. (a-\-x)(a — x — h). 6. (x—y-\-c—d)(x-^—c-\-d). 14. (a — x)(a — x-\-V). — r. (2+a;+ar^)(2~a;~a;2). 15. (a;-y)(3a: + 3y-2). a (a + 5)(a-6-l). 16. a:(a;2+l)(a;+l). ANSWERS. 17 17. (ax-l)(a^a^--ax-l). 18. (:r+3y)(a:-3y)(3.r-2y). 19. (2x-l)(2a^ + a^-]-l). 21. (x'-\-iJ')(x*-x'y'-{-y*). 22. ls + xXS-x)(9 + Sx-i-x^)(9-3xr''^y 83. ^(x' + y^a^-xY + y'). 24. c(a2 + c2)(a + c)(a-c). 27. 2a^(a; + y)(a7- Sy). 25. (a; + 7)(:r-3). 28. (2 a -5)1 26. 3(a-2^))(a-55). 29. 4(2a;-5y)2. 30. :i:3y2^Q^_|_5^^)(6^_5j^) 31. (Sxf-bzy. 33. (:r-y-z)2. 32. x(Ax'+l)(2x+l)(2x-l). 34. (a-3^>)(a+26-4). 35. (a; + y + 2)(a; + y-3). 36. (a^ + 2ab + b^ + c^) (a+ b + c) (a + b ~ c). 37. (a; - 3y) (a; + 2y - 4). 40. (x + 13) (a; + 7). 38. {l + x^)(l — x). 41. (x — y)(x — z)(z — y). 39. (3:i:-2y)(:?;-32/). 42. (:r + 3) (a: - 8). 43. (x-{-y-z)(x — T/+z)(x + 7/ + z)(x — i/ — z). 44. 5a;(:iry+4z)(a:2/— 3z). 45. (x^ -{-1) (Sx — 1). 46. (x — m -}- n) (x — 7n — n). 417. la + b + c)(a-\-b-c)(c+a-b)(c-a + b). 48. a^a'+l). 55. :t-2(35+^)(25-3^). 49. (1-7A)2. 56. (a -by. 50. (y + 9)(y-13). 57. (x + yf. 51. (:r + 15) (57 — 9). 58. ab(a — b). 52. (2a-3H2c)(2a-35-2c). 59. 32/2(a;-3/)(3a;+2y). 53. la + ^c)(a-Sc+6b). 60. (3a; + 2y)(2a;4- 3y). 54. (3a:+2y-z)(3:r-2y+z). 61. P(Sa+4:b)(2a-Sb). 62. (a + 2^'-3c + c?)(a-2Z> + 36' + c^). 63. (x' + 4:f)(x-2y). 68. (:^; + y _ z)(:i;-y+2 + l). 64. (2aa;-5)(2aa;-3^»). 69. 2(x + 7/)(x-{-y + a). 65. (3a:-2y)(6a;-4y+3). 70. 4:b(2a+Sc)(2a+c). 66. 2(a:+3y)(a;— 2y+3z). 71. (m + n)(w - w) (^ - ^). 67. (:r+y+l)(l-a;)(y-l). 72. 2a(3a; + y)(2:r- 3y). 18 ALGEBRA. 73. 2x(x+1){x - 5). 75. U(2x-y){^a^-{-2xy-\-fy 74. 2xl2a-x){4:aj'-\-2ax-\-a^). IQ. x{l-^x){l + ?>x+^x'). 77. {^^J^f)(^x + y){x-y)(x'~^f-^y% {a? — xy -\- y^) {x^ -\-xy-\- y^). 78. llm + lln){lm-\ln). 80. (2>z'+2)(4:z'-'S). 79. (4+92/2)(2+3y)(2-32/). 81. {x^+l){x~l). 82. (x + y+l){x-y + l). 83. (7a — 75 + 8m — 8w) (7a -75 -8m + 8 w). g4 [ {a-\-h + c){a-\-h-c) \ j{c + a-h){c-a + h)\ ' \ 2ah J r 2ah J ' 85. (a: -8) (a; -45). 87. {2h - e){a + h - c). 86. (:i:+2)(^-2)(a;-22/ + l). 88. 5 rr (5 ^ + 7y2)2. 89. a{a + h) {a^ + a5 + 5^) {a^ -ab-\- h^). 90. 2;r (a + c) (a - c) {p? -ac-\- c"). 91. (dx + 2y-Sz)(2x-Sy~\-Sz). 92. (4a:-y+2;)(a;-2y-2). 94. (a:- l)(a;3-a:2_3)^ 93. (Sa-b + 2c)(a-2b + c). 95. (5:?;-3y)(^-y-l). 96. (a + 2h-^c-d)(a~2b-{-Sc-d). 97. (:r + 2) (a; - 3) (x + 4:)(x- 5). EXEECISE XL VII. 1. 18a5c^. 9. 7a; -4. 17. 5y. 2. ITp^. 10. 2a^xy(Sx-y). 18. a;(:r + l), 3. 4a:2^V. 11. 2abc. 19. 3(a;-l), 4. 30xy. 12. a; - 3. 20. 2(a-b). 5. a — 5. 13. 2 (a + 5). 21. ^ + 3/. 6. a— a;. 14. 6^(a;-y). 22. x-y. 7. a + a;. 15. Sx(x + 4:). 23. ^-1. 8. 3a; +1. 16. c(a — 5). Exercise XLVIII. 1. 5x-l. 4. (3a;+2)(3^-2). 7. 2:r-5. 2. a; + l. 5. 35;(3:r2-^ + l). 8. a (a — 2a;). 3. a(3a-l). 6. 2(a;-2). 9. 2a^-Sx^^ ANSWERS. 19 10. 2a; -7. 15. 3 (a; + 3). 20. ^r^-S^ + l. 11. 5x(6x-l). 16. x^i-x7/+f. 21. 7:r+l. 12. y(3:r — 4y). 17. a -2. 22. yi^s^-xy-'bf). 13. 3rr-5. 18. 2(22/ + 5). 23. 2a<-^-4^4-a7-l. 14. 4:c + l. 19. 3a;(2a;2_5) 24. :?;-2a. Exercise XLIX. 1. :rfl. 5. y2_2y + 5. 9. 2 (a; + 2/). 2. y — 1. 6. :r— 1. 10. a? — xy -\- 'i/^ . 4. a; -2. 8. a; + l. Exercise L. 1. 12 aV. 15. (a7 + y)2(a;-y)^. 2. naa^y\ 16. 12(a;-y)2(rc3 + 3/3>) 3. a;2(a + ^). 17. I20xy {x + y)(x~y). 4. :r (:r + 1) (x - 1). 18. (a; + 2) (a; + 3) (:r + 4). 5. ala-{-h){a~h). 19. (a- 3) (a + 4) (a- 5). 6. (2a: + 1) (2^ - 1). 20. {x + 5) {x + 6) (a: + 7). 7. a^ + h\ 21. (a; + 2) (a; - 2) {x - 11). 8. (a;2+l)(a;+l)(a;-l). 22. 20o?h{a-h)(a-2b){a + U). 9. a;(r^+l)(^-l). 23. 240(a;+l)(a;-l)(a:+2)(a;-2). 10. xlx + V){3?-l). 24. 12a;2^(a; + y)2(a:-y)2. 11. (2«-l)(8a3 + l). 25. {a-h){h-c){c-a). 12. (a -5) (a + 5)1 26. (a-h){a-c){h-c). 13. 4(l + a;)(l-aO. 27. T'{x-l){x-2,){x-\-4:). 14. a;3_i 28. 12a:y(a;-y)^ 29. {a+h-\-c+d){a+h-c-d){a-{-c-h-d){a-^d-h-c). 30. 60 (a; -3) (a; -2)1 Exercise LI. 1. (2a;+l)(3a;-2)(7a:-l). 3. (a;-12)(a:-3)(a:2__2)(^_j_3^_j_9) 2.(a; + l)(a;-l)(a; + 3)(2a;+l)(3a;-2). 20 ALGEBRA. 5. 2(3:r-22/)(3a; + 5y)(4a; + 32/). 6. 2x(x'+l)(x-l)(x'-x-l). 7. 12(2x + l)(2x-l)(Sx-{-2)(x-S)(x-A). 8. (x-l)(x-2)(x-S)(x-4:). 10. (x + y)(x-y)(x + 2^)(x~2y). 12. (1 ~ a)^ 13. (a + b + c)(a- b + c^^ai- b - c){b + c - a). 14. c(3c2 + y2)(4c4_3y)(^_2/). 15. (m + 3)(m2-3m+l)(m^ + l). 16. (5n2-l)(4w2 + l)(5?z2 + n+l). 17. (5 - 1) (b' -b^- 8)(4 6^-8 6+1). 18. 6 r (r + 1)' (r - 1)^ Exercise LII. ^^ a^ + 5a + 5 ^i. ^' + ^' 4a; a^ + a — 2 a 2. ^-=±. 12. 5^. 22. -^. ^+i ,3 ^-2^ + 2 3(a + ^) * a;-7 * o?-2 ' ^ab(a-b) 4. ^.^-.. + 1. 14. .^^-^^ „ ■ 24. ^L+A+f. 4^+6aa;+9a2 a 5 ^!±^. 15 ^^±2^. 25 ^(^^-y) :r3-3/3 • Sa + 2b ' y{Zx^2y) ^ a^—a+1 ^g a — b — c ^ a + b + c-j-d a^-{-a-r-l a-^b — c a — b-\-c-\-d rj a^ j^ ^ + ^^-2 ^ Sx + 2 a-3 ' 2a;2_^2a;+l * 4^ + 5 8. ^Z:2 ^^ x_-d ^^ ^-xy^f ^ ^ + 4 x-\-\ (^ — y) 9. ^.nl. 19. ^Zl^. 29. :^2_|_^. 10. ^+^y-,< 20. <^^~^)' ■ 30. ^+^. a^—xy—'if' 0:^ — 3:^+1 a — b ANSWERS. 21 Exercise LIII. 1:^ 1. x — 1, 6. 2a — 3a; 5 a — X 2. ?>x~lO-\ — ~ 7. 12a; + 3+ "'•^ a; + 4 4^ — 1 3. 3a;-6 + ^^. a2:^; + 3-^ ^^ ^ + 4 x — 4: 4. a-2:. + -^- 9. a+5+-2^. a4~^ ^ — " 5. 2a;+6 + ^5-. 10. ^^-l + ^^ili-^. ^ — 3 5^ + 4^7 — 1 Exercise LIV. , 2y ^ ax-\-2> _ 2a^-ah~¥ 1. ^— . 8. -r— 15. — x-\-y 2a a + o 2. ^£L. 9. -2^. 16. 3^ + 2^ + V x-\-y a — h x-{-4: 3. ?!iii. 10. n^s. 17. ^+1. a; a-{-b x—\ . 2{a^-ax^a?) „ a?-2xy-f .^ x(3^-2x'-Z) _ a—x x-\-y x — 2 g 2(lla'^-20aZ)+8^^) ^^ 29a_ ^^ a^ + 2rg« 5a-66 " ' 4 ' * a + 2a;' 6. ■^^. 13. ^' 20. ^+^y+y^ . * a-f 5 a + 1 oj + a ^^ 2(19a-23a^-l) ^^^ ar 5-6a a;-3 Exercise LV. 9 a; -21 4rc-9 3 48a^-60ag 15a- IQg 18 ' 18 ' ' rna^c ' 60a*d? 4a7-8.?/ 3:r^-8a:y ^ 5 + 5a: _6 22 ALGEBRA. h~ {a -—h)(a — c) {h — c) {a — b) (a — c)(b-~ c) ^x^{a — h) xy 7 30(4a; + l) 5(2:r-l) S(Bx + 2) ' 15(a;-2)' 15(a;-2)' 15(a;-2y ^ an — bmn mnx cm — bmn TTinx Exercise LVI. ^ 163^+55a:+4a;.y-55y ^ ll/-7a;^-4a?y-8a;^y' 50a7 ' a^'i^ 21x'-2x'y-l^xy~2^'if ^ W(?-2ab^c-\-2a%c-o?d' 12 a^ ' ' a%''(? 180a^+54a^>-20a5^+331^>^ 8. 6a^~3a^-2 ^ 80a;«+64a;^+84a;+45 ^ a^^> + ^>^g + ag^ 60^^ ' ' abc 32^ + 9.y. ^,^ ___!,. 42 6y20 Exercise LVII. 2a:-l 3 2 2a:-.v • ar^-a;-30" " l-x" * (^-y/ :ir'-10^ + 21 \-\-x a^-o?b-ah^~b^ 7. a^> (a^ _ b"") Ix-VJ g 2a;« 2a-35 Exercise LVIII. V _2_. " 2. -i^. 3. ?^±^. 4. ^±^. 1-a l-a;* 1+r^ 3/ AKSWEES. ^2S ' :r3-95;2_|_26a;-24 ' * (x - af 7. — ^ 8. 0. 9. 2. 10. ^ . 11 0. ' (a-a;)(a + 2a;) ' xy{x. — y) 14 9a-105+^ 16 ?-(^-^) 17. 22;«-5^V+lQa;?/ + 5y^ (^ + 2/)'(^-y)' ' {a-\-h-\-c){a-\-h-c){a-\-c-h) ' a'a^-by Exercise LIX. 1. ^_. 3. -1^. • 5. 0. x — 7/ ar — l 2 _1_. 4 e+y . 6 1 '2 + ^ • 3(l_f_2/)(l-_3^) • (^ + a)(^ + ^,) a« + 2a'Z>-2a^>' + ^^ 8 ^'~^^ 9. —5^ — 10. 0. 11. 0. 12. 0. 13. 0. 2 + x 9a—10b + c ,^ 1 14. -— — — -• 15 (a — h) (a — c) (h — c) ' z{z — x) (y — z) Exercise LX. 1. «^. 4. i^. 7. _M£. 10. ^^. * 6c? ' 5Z> '42; * a^ 2.9a.. 5.i2^. a3=f/. 11.;^, 9 ax ^jpf {a — by 3. *• a — a.r. 9. 12. -^^ ^-^ ^ 4^2' ^ 24 ALGEBRA. 13. ^1 la -^^ 23. ^. 14 (^+^)(^'+Q^^+^) 19 ^ + ^ od g-« + ^ 15. 2a^(^-y). 20. 1. 25. ^-^+^ ^ x-\-a— h to oh , ^^- ,q:4^2- 21. i. 26. 1. 17. ■^^~^)^^~^). 22. -^^. 27 "^-y-^ ^ x-y ' x-\-y-\-z EXEECISE LXI. 1. 1. 5 («-^)(^+^)^(a^+:r^) o l±f 2. ^i:i. 6. V 10 :r4-l 3. _!-. 7. ^±i. 11 1±^ a: 4-1 2a; '1 + 07 ,_ (. + «)(x-«) 3_ 1 _ ^ j_ la ^±^ 14. (^+y )!. a^ — 2xy-\-'i^ 2y ^^ a{a'-\-2ah-\-2b') ^^ 1 ^ + a ' (a + ^)^ ' c{a—h — c) 16. a;. 18. m - 1. 20. ^(^~^) . 7-4a: Exercise LXII. 1. ^^^. 4. ^ 7. 1 10 ^^-^ . ^ + 8 2(a; + lf ■""• a + 5 2. 1|. 5. a;. 8. -/^. 11. ^. 3. 46. fi. 0. 8. 2. 15. «^ ANSWERS. 26 12. ^ + 3:r + 3-- + i. 13. 1. 14. 0. 15. a. 16. i. 20. a. 24. '^'[tT^'^^ 17. -^ — ! ! — -• 21. a{2a—x—y). 25. , — -• oc-\-ac—ao {x—z){y—z) 4 1 18. m. 22. . _ • 26. 3(a; + l) c(a-c)(6-c?) 19. (^L±k±Af.. 23. ^. 27. -^S: 2hc 2 (^ + 1)' Exercise LXIII. 1. a; = 16. 6. ^ = — |. 11. x = '^. 16. a; = 2. 2. a; = 5. 7. a; = 9. 12. a; = 10. 17. a: = 7. 3. a;=i 8. a; = li. 13. rr = 6. 18. r^ = 9. 4. :r = 1. 9. X = 7. 14. X = 12. 19. ^ = 9. 5. 37 = 8. 10. :i: = — 6. 15. 37 = 5. Exercise LXIV. 1.37 = 8. 3.37 = 4. 5. 37=1^ 7.37 = 2. 2. 3r = 9. 4. 37 = 20. 6. 37 = 4. 8. 37 = 3. 9. 37 =-2. 10. 37=11. Exercise LXV. 1. 37 = C. 6. 37 = 2. 11. 37 = !-— • 3 + 4a 2,^^3c-2a 7.37 = 0. 12.3^ = 6-1. bh~c a .r= , '-{ ■ a x=-^. 13.^=0*. 4. 37=1. 9. 37=— 14. x = bm. a-b 2a + b 26 ALGEBUA. 16. .= 24. 17. ^=2rJ:. — 1 h la 37 = --- -. 19. x = - -. 20. a; = — o — (f bc-{-d Exercise LXVI. 1. a;=7. 7. a: = 8. 13. 07=7. 19. a7 = 3f. 2. a; = f a a;=-14. 14. a;=a+-^. 20. x ""^ a-\-b ' a-\-b 3. a? = -||. 9.07=14. 15.07 = ^. 21. o; = 12. 4.ar=li. 10.07 = 111. 16. o: = 0. 22.07 = 3. 5.07 = 3. 11.07 = 2. 17.07=1. 23. o: = -9. a 07 = 0. 12. a; = — 2. 18. 07 = 1. 24. 07 = |f. Exercise LXVII. 1. 24. a 80. 5. 28, 32. 7. 76, 24. 9. 3456, 2304. 2. 168. 4. 8. 6. 2f , 47-f. 8. 960. 10. 36, 9. 11. 420. 12. 85 gallons wine, 35 gallons water. 13. 18 pounds saltpeter, 3 pounds sulphur, 3 pounds char- coal. 14. 28,18. 15. House, $600; garden, $250. 16. 17200. 18. 47, 23. 21. 19, 20, 21, 22. ^^ a + b a-b 19. 7, 32. 22. 42 years. 2 ' 2 ' 20. 112, 96. 23. 20 years. 24. Father, 36 years ; son, 12 years. 27. 2 days. 25. A, 28 years ; B, 14 years. 28. 1-^ days. 26. Father, 40 years ; son, 10 years. 29. 26|- days. 30. A, 168 days ; B, 84 days; 0, 42 days. 31. 10| days. 32. B and C, 21^ hours ; A, B, and C, lO^f hours. 33. 10 days. 35. A, 37| minutes ; B, 25 minutes. 34. 40 minutes. 36. 4|- hours. 37. 48 minutes. Oft Q^^g 39. 48f minutes. 41. 7 miles. ' db + ac-{-bc 40. 42 hours. ANSWERS. 27 40 ^g 44. 1st, 5 miles per hour ; 2d, 3 miles per b-i-c hour. 43. 160 miles. 45. 175 miles. 46. I. 16^ minutes past three ; II. 32^ minutes past six ; III. 49-^ minutes past nine. 47. I. 32y\ minutes past three ; II. 5y\ and 38y\ minutes past four ; III. 21-^ and 54^^ minutes past 7. 48. I. 38^^ minutes past one ; II. 54^^ minutes past four ; III. 10-^ minutes past eight. 49. 5^^ minutes past two. 50. 700 leaps. 51. Greyhound, 450 leaps ; hare, 600 leaps. 52. 240 leaps. 53. Length, 13 feet; breadth, 8 feet. 54. 56J square feet. 55. Length, 27 in. ; breadth, 14 in. 56. Length, 15 feet; breadth, 11 feet. 57. Ill pounds tin ; 69 pounds lead. 59. 189. 58. 76 pounds gold ; 30 pounds silver. 60. 6J ounces. 61. 330 feet. 62. $228f . 63. 24 shots. 64. 60. 65. 12,000 square yards ; 56J cents per square yard. 66. 20 seconds. 68. 103 sheep. 70. 10 gallons from 1st cask, 67. $55,500. 69. $3700. 4 gallons from 2d cask. 71. 60. 72. 1504. 73. .39. 74. 9f miles. 75. ^Zli. 76. ^^^^ of 1st kind; ^^^ of 2d kind. b-]-c h~a o—a Exercise LXVIII. 1. x = 2. 4. a: = 12. 7. a; = 29. 10. a: = 3. y = L y = -3. y = 23. y = 2. 2. a; = 2. 5. rr = 6. 3. a; = 8. 11. a; = 8. y = ~l. 2/ = 7. y = 4. 2/ = 20. 3. ^ = 2. 6. a; = 7. 9. :r = 7. 12. a; = — 3. y = 8. y = 9. y=14. 3/ = -2. 13. a; = — 2. lA. x = 7. ALGEBRA. Exercise LXIX. 1. a; =10. 4. a: = 4. 7. 07 = 20. 10. 07 -= 2. y = 7. y=l. y = 10. y = 2. 2. 0^ = 17. 5. 37=5. 8. 07 = 2. 11. 07 = 2. 2/ =19. y = 5. y = -3. y = 3. 3. ar = 2. 6. 27=21. 9. 07 = 8. 12. 07 = 9. 2^=13. y=12. y = 4. y = ~3. 13. ^ = - -5. 14. 07 = --h y = 14. y = =-i Exercise LXX. 1. x = S. 4. x = 2. 7. 07=12. 10. 07 = 4. y = 3. y=13. y = 3. y = 3. 2. ^ = 6. 5. 07 = 8. 8. 07 = 2. 11. 07=-:~2. y-3. y = l. y = e>. y = 19. 3. ar = 6. 6. x = ^. 9. 07 = 3. 12. 07 = 7. y = 8. y = 4. y = 5. y = -5. 13. X = 2 > 14. 07 = 11. y ^ - -3. y-- -10. Exercise LXXI. 1. x = l. 7. 07=11. 13. 07=12. 19. 07 = 4. y=7. 2/ = 6. y = 6. y = -l. 2. ^ = 2. 8. 07 = 3. 14. 07 = 3. 20. 07=^2. y = 3. y = 2. y = 2. y = 3. 3. a: = l. 9. 07=7. 15. 07 = 9. 21. 07 = 6. y = 8. y = 5. y = 8. y = 5. 4. x = 9. 10. 07 = — 5^. 16. 07 = 5. 22. 07=19L y = 8. y = -9i- y = 7. y = -17. 5. a; = -3. 11. 07=7. 17. 07 = 7. 23. 07 = 4. y = -2. y = 8. y = 2. y = 3. 6. x=1. 12. 07 = 14. 18. 07 = 9-^. 24. 07=7. y = 10. y = 46. 25. 07 y y = 2. = 21. = 20. y = 9. ANSWERS. 29 Exercise LXXII. 1. ^=^. a x=\. 2 aJ a-b „ 1 2. x = -^ — -■ 3a «?-*? 3^-V^ + d ar — cp y = — ' — • aq~op 3. a; = 4. a; 3/ hn-aq ^^' ^ ^_5' np — mq ^ ^ ap — hm ~ ^ ~ a + 5' ^^-^^' 11. x^a-h. ce~hd y = a-\-h. "^^■^^^ 12. x^ «^ ^. ' (a-l)(^-l) ^—c a— I y 13. X h. X J- . , 7 ??^n' + '^^'w y : 2 a — h 2 a/-6c^ 14. 2: a c{ac-d) ^ ^ 2_52 7 ^==_^^!^ 15. x^{a + hf. h^c-df y^{a-h)\ J^hc_^ 16. x=a-\-c. 30 ALGEBRA. ,„ ah(? 19. x=h — a. 17. x=a r- . ac-\-b y = a — o. y ^ ah(a + h) V^^ 20. x^= — ^^ — ' — ^• 18. x-=a-\-'b. _ a?-h^ y = a-h, ^ 2(a2 + 62y Exercise LXXIII. 1 5. x = ~\. 1 2. a; = :; — V = — 3. a; = 6. a + ^ -^ c ^ m a^ + h^ 9. :r— ' am-\-hn hm — an Exercise LXXIV. 1. 37 = 2. 4. :r = 5. 7. a; = 20. 10. 37 = 4. 2/ = 2. 2 = 2. y = 6. z = 7. y = 10. 2 = 6. y = 0. 2 = 6. 2. x = b. 5. x==l. a a: = |. 11. 37 = 6. y = 6. 2 = 8. y = 4. 2 = 6. 2 = 36i y = — 6. 2 = 5. 3. a:=l. 6. x = 2. 9. x==2. 12. 37 = 46. y = 2. 2 = 3. 2=10. y = 3. 2 = 1. y = -2I 2 = 1. ANSWERS. 31 13. x=l. y = 2. z = -3. 15. a: = — 1^. y = 2i z = 6i. 17. ^ = i 19. 37=6. y = -12. 2=18. 14. x = l. y = -l. z = 2. 16. :?; = -4. y = -3f. 2 = -4i 18. x = 9. y=ll. 2 = 13. 20. 07 = 1. 0.1 ^ 2 oo ._^^+^-«^ a + 6 25c .. y = ! a + c . 2ac 2 _a^+&"-g^ 2a6 25. x=lh y = -3i 2=2,1,. 27. 07 = 2 = = 17. = 22. = -25. _(a+5 ~c)(a+h) a- i-b-\-c b+c 23. x=i. 24. 37= 3J. y = i y=if 2=i. ,2 = 1^. 26. 3. = ^+^. 2a _ ah-\-ac-\-bc-h^-2^ ^ 2{¥-c') 2>bc ~- ab — ac — b^ ^■~ 20)^-0^) rto ' / I z.\ (a—b—c)(a-{-b) 28. 37 = — (a+o). v = ^^ -^ — ■ — ^• Exercise LXXV. 1. 41, 7. 2. 35, 11, 98. 3. 6, 4. 4. 4.78, 1.0095, 5. Father, 35 years; son, 14 years. 6. Father, 40 years ; son, 10 years. 7. A, 1 116; B, $166. 8. Wheat, %1\ per bushel; barley, $f per bushel. 9. A, $31 ; B, $27. 10. Horse, $100 ; cow, $50. 11. -f. 14. -^. 17. 28. 20. 75. 23. 717. 12. yV- 15. I, f . 18. 24. 21. 36. 24. 759. 13. |4. 16. If. 19. 84. 22. 69. 25. 126. 27. 7 i hours, 4i hours. 28. 3 miles per hour. 26. 432. 32 ALGEBRA. 29. Distance, 20 miles ; rate per hour, 8 miies. 30. Time rowing down, 4 hours ; time rowing up, 6 hours ; rate of stream per hour, -f miles. 31. 20 pounds @ 42 cents ; 10 pounds @ 54 cents. 32. 40 pounds @ 90 cents ; 90 pounds @ 28 cents. 33. Eye, 20 bushels ; wheat, 52 bushels. 34. A, $3t; B, |3; 0, $2i 35. A, If hours ; B, 3 J hours ; 0, 7 hours. 36. 54 boxes ; 36 bales. 37. 1st, 36 days ; 2d, 45 days 38. 21 crowns ; 63 guineas. 39. A, 20 days ; B, 30 days ; 0, 60 days. 40. A, ^-^' minutes; ^^' ^^^^^ ® ^^- ac + hc-ab $5000 @ 4%. minutes; 42. $20,000 @ 5^ minutes. 43. $25,000 @ 6%. B, ^ ^f^^ minutes; 42. $20,000 @ 5% ab -^ DC — ac 2abc ab-\-ac — be 44. Sum, ""-^"^ dollars; rate, M%j} <^,, n — m an — bm 45. Sum, — dollars ; rate, y~ — ^ % . a — b ad — be 46. 1st, $3000® 4%; 2d, $4000® 5%; 3d, $4500® 6%. 47. $6750 ® 3% ; $6000 @ 5%. 48. 3 per cents ® $60 ; 4 per cents ® $75. 49. 3 per cents ® $75 ; 3^ per cents ® $88. 50. $2000 ® 2% ; $500 ® 4%. 51, 2660 square yards. 52. Length, 14 feet ; breadth, 10 feet. 53. Length, 90 yards ; breadth, 60 yards. 54. A, 9 miles ; B, 8-j^ miles. 55. A, 12 miles ; B, 10 miles. 56. Express-train, 45 miles ; accommodation-train, 30 miles. 57. 315 miles. 60. Ox,$105; sheep, $8|; lamb, $5i. 58. 30 miles. 61. 2 horses, 4 oxen, 94 sheep. 59. 33i miles per hour. 62. A, $50; B, $30; C,$20. 63. A,$9f; B,$5i; C,$3. 64. A, $3i • B,$2A;C,$I. ANSWERS. 33 Exercise LXXVI 3. a;V- 4. ^ 6. :^-^e>x'-{-12x + ^. 7. :r* - 80.-3 + 24a:2_ 32-^ + & a;^ +15a;^ 4- 90a;3 + 2700^^4- 405a: 4- 243. l.a« 2..- 3..y. ..-':^. 5.?^^ a a;«+15a;^ + 90a;3 + 2700^^ + 4050: + 243, 9. 1 + 10o: + 40^:2 _^80o:3 + 80 o:* + 32o:«. 10. Bm^ - 12m^+67n - 1. 12. - 125aV3/«. 11. lea^SV^. 13. 49mVo:y. 32o:V 15. 81a:*+108o.'3+54or^+12a: + l. * 243a«^>V' 16. 16a:*-32ao:8 + 24aV-8A + a*. 17. 243 a^ + 810ax^ + 1080 aV + 720 aV + 240 a*a: + 32 a*. 18. 16o:*-32a:^y + 24a;2^-8a:/ + 3/*. 19. o;V - 12^/ + 60a;y - 160o:y + 240o;«y« -192o;yi + 64a:«y^. 20. a^o' - 21 a«6« + 189 a'b'' - 945 a^b* + 2835 a^J^ - 5103 a^b^ + 5103 aZ> - 2187. 21. -243a^«^V. 22. 729a:y». 23. -S125aWx''. 27. 8-36a:+102o;2_i7x^_|.204o:*-144ar' + 64a;«. 28. l-6a:+15a;2_20a;3^15^4_6^_|_^6 29. l-Sx + 6a^-7c(^ + 6x*-Sa^ + a^. 80. 1 + 4a: + 10:^2 +16a;3_f. 19^4 _|_ 15^ ^10:r« + 4a;^ + 0:8. Exercise LXXVII. 1. ±a^ ±A ±2a%, 4, ax'f, ±2a^bc', -2a\ 2. -12c2^V 15^»V, ±42c*0. 4. 2ai''^. 7. 170. •^ ' 72« " 32* 6. 92. 9. 1. Exercise LXXVIII. 1. a^ + 2a-~l. 3. 2a^ -Sa^x- ax^. 2. oc^ —xy-\-if. 4. 3a,-^ — 4o:?/^ - - 23/^. 34 ALGEBRA. 5. 2a* + 4aV -46'^ 10. 2-3a-a2^2a^ 6. 2a^-5x + S. 11. Sa-b + 6c-\-d. 7. 4.ar'-2ah-\-2b^. 12. 6a^- ^a^i/ -ixf + y^. 8. 3^-2ar^ + Sx~4:. 13. m*-2m3+3m*-4m+5. 9. a;3_2^y-f 2a:y2-3/^. 14. :^-\xy-f. 15. a;2._2^y + y2-^. 17. 1+? + ^ + ^. X x" ' a;» 16. |-|a^ + |. 18. ^-1 + ^. 19. a;2 + | o z a 2 Exercise LXXIX. 1. 347 ; 69.4 ; 43.21 ; 37.89 ; 8.008. 2. 129.63; 2.1319; .4937; .75416. 3. .9486 ; 2.4919 ; .6557 ; .0923 ; 4.1231 ; 11.3578 ; 18.6348 4.119.5633 ; 1.5811 ; 44.7213 ; .5477 ; .1732 ; 10.5356 5. .0333 ; .0632 ; .0707 ; 1.4142 ; 2.2360 ; 1.8027 ; 2.9325 6. i; ^; I; H; «; i 7. .7071 ; .8164 ; .8660 ; .1767 ; .2338...,. ; .2190 ; .9258 ; .2886 Exercise LXXX. 1. aT + 2y. ^. a?^x^\. 9. \—x^x^—:i^. % a -3. 6. \-^x^^o?. 10. o?-^Zah-^W. 3.07 + 4. 7. a2-2a-l. \\. (?-^hc-\-W. 4. ^-ax-o?. 8. 4ar'+4a;-l. 12. 2a^-\-^ab-ZW. Exercise LXXXI. 1. 65. 4. 9.6. 7. 11.7. 10. 192. 13. 32.1. 2. 48. 5. 478. 8. 108. 11. 2.34. 14. 46.8. 8. 64, f, U4. 9, 1,4,1, 1J3, 3,8i J.5, 9.36, ANSWERS. 85 16. 20.53. 17. 1.111. 18. 27.55. 19. 4.5045. 20. 1.7479. 21. 1.3572 ; .5848 ; .2154 ; 1.6874 ; .7368 Exercise LXXXII. 1. 3o-5^>. 2. l-x + ar^. 3. 2 — x. 4. 3a;- 1. 5. 1-7/. Exercise LXXXIII. 1. ±7. 4. ±h 7. ±1. 10. ±2|. 2. ±3. 5. ±i. & ±5. 11. ±5. 3. ± V3. 6. ± VIl '. 9. ±3. 12. dz2. 13. ±2. 1.^ -^ *^.-i- L4. ±yj~: a ExEBOisE LXXXIV. 1. 2, - 6. 8. 1,-f 15. 3, 0. 22. 3, -f 2. 8, - 2. 9. 14, -f 16. 11, 0. 23. 4, - 1. 3. Hi i- 10. h i- 17. 3, - 1. 24. 3,-f 4. 8,-1. U. 6, - IJ. 18. lzfcV3. 25. 2|, 0. 5. 2i, -1. 12. 2, |. 19. -3,-3|. 26. 7, — 2. 6. i-i- 13. ± Via 20. 5, lif . 27. 5, 2. 7. 3, - 2. 14. i.-4f. 21. 10, - 29. 28. 3, 4. 29, . 5,^4f 30. 13, ± VS-^. Exercise LXXXV. 1. —a± a V2. 3 !^ _ T^^ 40 w 2. 2a±aVn. '2' 2 " ' ~2 g^ a(a + h) a{a-b) ^ c±:^(^ + ^ac ^ ^ a — h' a + h ' 2{a-\-h) ' ac' 3. a -. 9. '^^^ — <^~^c ^ 0. 10- - «> — ^• ' (t ' a + 6 — 2c? ' XL l±V^ 36 ALGEBRA. 12. a'-\-h^±(a- h) Va^ + h\ 14. 1, - a. ^^- 2"^ ' "2"' 16. ±V^. 1,. ,, _ ^_M:^. 18. 4., - f . 19. 6d.3V?+4 20.™, -^. 2i.2!±i!±V(^:zi!Z+4^. 22.il. n m^ 2a6 23. ±m Ji r. 35. m + w, ^ 24. l_ 3a^2aVa 36. 2a-5, a + 2h. {a-Vf ^ a±h a-h a-\-h a — h a — b a-\-h 'a-h' a + h 38. 2a + 65, a -85. 26. {in-\-7if, —{m — nf. ^ ^a-5b ^ a-2b 39. ^-^b, -(5a+b). Gab Sab 4a2 + 95 ac be 29. a + |, m-2a. ^^- "'' ^T^* 3ft-25 2ft-35 42. 1, Q ' 5 ' a+b+c a + b 43. 1 H^szll. 30. 31. 2 a, 2i c(a-—b) 32. a — b, c. 44.-1, _^~^^ 33. m±(n-p). 2a-2>b ^ 2 ^' a-25 a + 26 ExEECisE LXXXVI. 1. ±1. ±2. 7. 1,-1,1. _ 2. 1,2,-3,4. 8. i, K-l=fcV-3Y 3. 0, -i±V6. 9. -^, i(i ±Vi::3)^ 4. 0.5.-7. 10. ±1, i(dzl±V-3). 5. -2, 3, -4, 5. n. 0, a, ±5. 6. a, ^ (— a~3:i: V— ^c^-— oa-t-1). 12. -1, 1 ± -1 V -iiw'-4». ANSWERS. 37 Exercise LXXXVII. 1. ^'-3:r + 2 = 0. 5. 2a;^ + lU + 5 = 0. 2. a;'-4a;-21 = 0. 6. 63a:^-32a: - 63 = 0. 3. 6:^2- 5a; + 1-0. 7. 16a:* -153:i-'^ + 81 = 0. 4. (Jx' + 5x-6 = 0. 8. a:(ar^~6a:^ + lla;-6) = 0. 9. x{4.x'-1x-S) = 0. 10. a;'-4aar + 3a2-4a5-4^^ = 0. 11. x''i-ax — 6a''-\-5ab — h^ = 0. 12. a,** — a'ar — a: — a' + a = 0. 13. Rational and both positive. 16. Impossible. 14. Rational and of opposite signs. 17. Surds and botli 15. Rational and of opposite signs. negative. 18. Impossible. 19. Rational and of opposite signs. 20. Rational, equal in value, and both negative. Exercise LXXXVIII. 1. Max., 13. 4 ^^^ (a -by 6. Min., 4. 2. Min., 4a. * ' 4 7. Min., 8. 3. Min., ±2. 5. Max., =b^. „ ,4- 4a5 ' ,-- * 8. Mm., ; — 9. Max.,±Ki- . . ^" + *^ \a 11. Each 10 inches. 12. J. 10. Each 10 inches. Exercise LXXXIX. 1. 1,-2. 7. ±V2j, 2. zfcl, d=2. 8. ±5^±2. 14. VA±AV89. 3. ±3, rbf 9. -^-2,^^^. 15. 1,1,-3,5. 4. rbl,±i 10. 0,-J,-i 16.-1,2,-3,4. 5. 1, f 11. Impossible. 17. 1-J-, dr-J, f 6. ±2, ±i 12. 1, V-4. 18. i:l, ±3. 38 ALGEBRA. Exercise XO. 1. 10, 11, 12. 3. 1, 2, 3. 5. 6. 7. $lli 2. 3,4. 4. 12. 6. 12. 8. 31i- cents. 9. 12 inches. 14. 7.63 , 12.36 10. 121 yards, 120 yards. 15. A, 36 hours ; B, 45 hours. 11. 16, 16, 28. 16. 5 hours. 7 hours. 12. li yards. 17. 3 hours. 5 hours. 13. 2.4142 inches. 18. 8 feet ; 4^ pounds. 19. 39. 21. 5 miles per hour. 23. $80. 20. 15, 20. 22. 4550. 24. $60, 25. 100 shares® $75. or $40. 26. 15, 9, and 6 yards ; or, 2|, 1 J, and f yards. 27. 5f miles per hour. Exercise XCI. 1. 9, 4. 6. 20,-6. 11. 40, 9. 16.4,-21. 4,9. 6,-20. 9,40. 1,-li. 2. 25, 4. 7. 10, 2. 12. 6, 5. 17. 3, - 3|. 4,25. 2,10. 5,6. 4,-3if. 3. 22,-3. 8. 4, 2. 13. 10, 2. 18. i. 3, - 22. 2,4. 2,10. *. 4. 50, - 5. 9. i,f 14. 5,-3 . 19. i. 5, - 50. hh 3,-5 2. 5. 13, - 3. 10. 3, - Iff 15. 7,-4 20. ± V2|. 3.-13. -4,8ii, 4,-7 2=F V2j. Exercise XCII. 1. i 3, =F 8. 3. ± 4, ±3V3. 5. ± 7, zfc 5 Vi. ±5. ± 5, ± V3. ± 2, =F 9 Vi 2. ± 2, =b V|. 4. ± 5Vf. 6. rt 2, =fc V2. ±i +2 V|. v|. zt 4, _-t 3 V2. ANSWERS. 39 7. ± 3, ± 5VJ. 8. ± 2, ± 2f|. 9. ±3. la zb 3. d=2, ±V}. ±3, ±2^. ±4. ±1. Exercise XCIII. 1. 4, 2, 3 zt V2r. 3. 4, - 2, 0. 6. ± 3, =b 2. 2, 4, 3:=fV21, 2,-4,0. i 2, ± 3. 2. 3, 2, -3 ± V3. 4. 4, 2, ■|-(-13 ± V377). 6. 3, 2. 2,3,-3::fV3. 2,4,J(-13=fV377). 2,3. Exercise XCIV. 1.4,3. ^3^^^ 24. ±3, ±2. -3,-4. 2 ±2, d=:3. 2. ±5, =h7Vi. a^j 25. 3,1. ± 2, ± 3V|. 2 ■ 1, 3. 3. 8, - 3. 14. 3, - 2. 26. a, 0. If, - 4. 2, - 3. 0, - a. 4. d= 7, ± 1. 15. i 4, ± lOf. ^ ^^±1' ±1, ±7. ±9, ±:2|. * a-5 5. 2,-2^. 16.4,2. ^ 2ah -1, -2i|. 2,4. a-^> 6. 8, 1. 17. 10, - 10-J. 28. ± (a + h). -1,-8. 15,-161 ^{a-h). 7. ± f la 3, - 2. 29. ± 4, 0. ±f 2,-3. 0, ±4. a 2f , - If. 19. 12, 10. 30. ± 4, ± 3. H,-2f. 10,12. ±3, ±4. 9. ± 1. 20. |(ajrV2^>^-a^). 31. a + 5, 0. i-J. |(a+V252-a2). a + 5,0. 10. 4. 21. ±lf, =fciV3. i{a-6±V(a+36X«-&)}. _ 4. ± 1|, ifc^VS- H«-^TV(a+36)(a-6)}. 11. i(±3=fc V29). 22. ±6. 32. 5, f. j(4,3=fV29). ±3. 3,-1 J. 12. ii,d.ivs^ ""'H- ''•^^• 40 ALGEBRA. 34. 35. 3, 2,^(-3±- 2, 3, K-3h^- 37. 6, -3f 4, -3 A. 38. ± 36. 5,-2,i(3±V=67). 2,-5,K-3^V=67). /2a-3b\ I — 7 fl^) i- Exercise XOV. 1. 7,5. 3. i(3d=V5), J(1±V5). 2. 34, 43. f ±iV5, ■|^^iV5. 6. 15 miles; 3, 2^ and 4 miles. 7. 73, 37. 9. 88 yards, 55 yards. 11. 8 feet, 10 feet. 10. 290 yards. 12. $3000, 7% ; $3500, 6% 4. 16, 10 R 2 3 8. 16. 12. Exercise XCVI 1. 19, 8. 10. 8. 20. 91, 28. 1,3. 1. 9,72. 2. 1,4. 11. 21. 21. 8f , 23-i. 11,4. 12. 17i 70. 3. 5. 12. 49. . 22. i 4. 22. 1- 4. 9, 19. 13. 32. 23. 6, 12, 18, 24. 2,1. 21. 56, 42, 28, 14. 5. 15, 7. 14. 19. 8, 16, 24, 32. 2,5. 476. 24. 10. 6. 4, 9. 15. A, A. A. 1^- 14. 13,5. H.tV A, A- 176. 7. 6. 16. 8. 25. 149. 2. 17. 59. 26. 6, 13. 8. 28, 18, 8. 18. 3, 6. 17,7. 1, 8, 15. 21, 2. 27. 14 half-crowns. 9. 3. 16, 32. 9 4-penny-pieces. 5. 19. 301. ANSWERS. 41 29. 19 oxen. 30 34, 37, 40. 31. 905. 6Z. 34. 1 sheep. 13, 8, 3. 3620. 17. 80 hens. 1, 3, 5. 33. 14. Exercise XOVII. 8. (a^-{-b^){(f+d^). 9. m^ + l. ' 10. ia\a-b). Exercise XOVIII. 1. x^; x^ ; x^ ; a^ ; a^ ; a^ ; a^b^. 2. x^y^z] x^y^z^\ a^b^c; 5ab^c^ct^. 3. ^a^, -VaFb- ^V^' \ 2>V^\ 1 . ^ . 6 y . ^* . ^xf d^' x'lf' a^ ' ]/' ab^ 6. ^xyz"'^) x'^y'^^z; ab'^C^; or^b'^^-46 + a-^it. 14. 4 ; 10 ; 4 ; i ; ^. 16. Sa;-^ - 3^;-^^ + y- 15. at-3a5^ + 3aU3-5. 17. 2x-^l-^x-^. 18. 24. 20. x^. 22. a'"-\ 24. 1. 19. a^K 21. 2aUi + llZ>. 23. x^'i. 25. a;(«-i)i'--y(i'-i)« Exercise 0. 1. V45; Vl89; VSOO ; V^^; -V^. 2. ayV2^\ 2V6; ba'dVbd, 5. 10^; 2a:/-V^20^; 3mV-v/4^; 7a*i*\/4^». a (a-^.)^a; 5(a-5)V2. 7. 4ac-{/5^5¥;,42VIl^; 27y\/3^ ; 55 V6. 8. iV5; |V3; |V2; fV5^; ^. 9. ^-^2^?; i-v^; 25^8^6; ^-sJUcxy. la 2|V5 ; tI^ V21 ; —^ ; — - yt G 11. a5a;t; 2a^h^^x; Iba^h^y^. 14. 7.071065 ; 42.42639 ; 0.707107 ; 0.1414213. ANSWERS. 48 Exercise CI. 1. 3V7. 4. 24 V3; V6. 7. A\V30- 2. 6V7; 5VlO; 9VJ. 5. SJf V7; 2. 8. 4|. 3. 5^; 4^; S-\/b. 6. I-VS ; HV15. 9. Y^^. Exercise CII. 1. 3V2; 1^; 2\/3. 7. iv'225,000a^y^ 4. '•^^"*^^*- 8. «^,W. 3. 3a/7 ; 4V2 ; 2-\/22. ^V ^ ' 4. 3Vl9; 5^; 3^3. 9. 6 - 30 V3 + 24 VlO. 5. "lax-ypn:^^ \ a^fJ^\ 10. ^; (2a5)* a" ' Exercise GUI. 1. 29V3; 30VI0+164V2. 6. {a+l(dh)Vab. 2. 13^/2 ; 33^/2. 7. ahcVW?. 3. 48 V2 ; 0. 8. 0. 4. |V3; ftVT5. 9. 8a2V3^; 9^. 11. -^i Va-80a^>V^+80//Va-32^/V/>. 17. 64 a^-96 a» VS+60 a»-20 a^V^+3f a«-f aVa4-^ ; _ a 64 16 64a;V''-192aJV*-v/^+240:rV"'^y^ -160a;V» 4- 60 a;*^/^ - 12 ^2/2^_^^^ ^ 2 ^ 16 19. a^^-^d^tc^-t - 3 a%"^ + 3 a^»-»c-i(^ t - c~^d^ ; a«" - 4a* + 6 - 4 a"" -\- a'^" ; 81 " 20. -f-r ^-f 5tct Z)2^^ 3^3 ^5 27a«^>3' at 3a2 . 3 b^ ^-^+¥«^^*-27i^; 21. x^^ + 2>x'^y^ + 7f\ M 2^>H 4a^Z>* Sa^ 22. l^2x-^-2>xr^-]-4:x-\ Exercise CIV. 1. f(V7-V5); |(2V5+V6); 5V2-6: 30 + 12V6. 5 — 2Z>-9a «• U-l=tV5)' K-1±W53)» 10. 6^^. 12. iff 2a 2a 5. 1, -1 J(li-^V4T). 11. 4,1. 13. i(l±V=^), ±V=^. 14. 5,-2^, i(5±V97). 15. J,-|,|(1±V33). 16. 4a2+5* 86^ 17. t 2i0. 18. ± 2 a/2, d= i V=^, ± i^(5 ± V41)'. 19. - 8i, 5, K4 =F 3 Vl09). 6f , - 2, f (- 3 ± Vl09). 20. i(a + h + l)^iV^^^+l=PiVU + l. |(a - 5) =F i V4M^ ± i V4rfl. 21. ±2, ±1, |(±V6zhV2). ±1, ±2, |(±V6^V2). 22. 6, 2, K-QiVSI). 2, 6, |-(-9=fV33). 23. l-(a2 + ^,2) i i-Vl0a2^2_ 3^4_ 3^4_ 43 45 1 (^2 ^ J2^) ::p i-VlOa^J^ - 3 a*- 3 ^>^ 4b 46 24. 3, 2, J(5d=V6l), -liVlO, _1±V^. -2, -3, i(-5±V6l), IztVlO, 1±V^. 25. 3 zb 2V2, 0. (3 ± 2 V2)2, 0. 26. 6, - 4i 12, - 9. 27. ± 16, ±4. rh 4. =fc 16 9a ^. ±2 28. ?^(^a±-Va^ + a), ^(a±Va2 + 9a). V 30. 9, 0. T\.r* O TY JliXVO. 1 1 Exercise CIX. 1. 0.7781. 12. 2.6232. 23. 0.0970. 2. 1.1761. 13. 1.0791. 24. 1.5741. 3. 1.3222. 14. 1.7781. 25. 0.3222. 4. 1.1461. 15. 1.8751. 26. 1.2040. 5. 1.5441. 16. 0.8751. 27. 8.7481 - 10. 6. 0.9542. 17. 8.3222 - 10. 28. 9.7993 - 10. 7. 0.9030. 18. 9.5441-10. 29. 0.2431. a 1.6902. 19. 7.5441-10. 30. 2.0212. 9. 1.3980. 20. 7.6020-10. 31. 8.0212-10. 10. 1.4771. 21. 8.6990-10. 32. 0.0212. 11. 1.6232. 22. 1.0970. Exercise CX. 1. 0.9030. 7. 0.1003. 13. 0.2258. 19. 2.9579. 2. 1.3980. 8. 0.3495. 14. 0.4660. 20. 0.6361. 3. 3.3804. 9. 0.0596. 15. 0.2045. 21. 2.4465. 4. 3.8168. 10. 0.1690. 16. 0.2415. 22. 0.4730. 5. 2.5353. 11. 0.1398. 17. 1.1650. 23. 0.5243. 6. 3.4950. 12. 0.0768. 18. 0.3904. 24. 1.3280. 25. 1.1569. Exercise CXI. 1. 9.6020- -10. 11. 0.1761. 21 . 0.4559. 2. 9.4559- -10. 12. 1.1461. 22 . 8.8539-10. 3. 9.7781 - -10. 13. 8.2219-10. 23 . 7.6320-10. 4. 9.6320- -10. 14. 7.3980-10. 24 . 7.5441-10. 5. 9.8539- -10. 15. 8.1461 - 10. 25 . 5.1707-10. 6. 0.1461. 16. 1.8539. 26 . 4.8293. 7. 0.2219. 17. 2.6320. 27 . 5.9333. 8. 0.3980. 18. 6.6320-10. 28 . 4.1040. 9. 0.3680. 19. 1.2219. 29 , 2.8627 - 10. 10, 0,5441. ^20. 9.544X - 10, 30 . 7.1375. iS ALGEBRA. 1. 1.7782. 2. 2.0043. 3. 2.9996. 4. 3.9957. 5. 3.7329. 6. 3.5775. 7. 4.7350. a 4.9581. 9. 4.0000. 10. 4.0004. 11. 4.8490. 12. 4.0802. 13. 7.9943- 14. 9.9430-10. 15. 0.0000. 16. 2.9430. 17. 1.8686. 18. 0.8494. 19. 8.9528 - 10. 20. 1.9990. 21. 17,730. 22. 4295. 23. 339.6. 24. 88.8. 25. 53,910. 26. 93.98. 27. 0.7586. 28 1.9. 28. 0.1099. Exercise CXII. 30. 0.695. 31. 0.001051. 32. 0.0455. 33. 41.64. 34. 0.02989. 35. 0.3131. 36. 8.294. 60. 61. 62. 63. 64. 65. 66. 67. 37. -28.47. 38. -0.3144. 39. 0.0104. 68. 40. -660,600. 69. 41. 1.308. 70. 10. 42. 0.6307. 71. 43. 105.4. 72. 44. 0.7167. 7a 45. -111.3. 74. 46. 0.001086. 75. 47. 0.7277. 76. 48. 0.2084. 77. 49. 6.963. 78. 50. -5.219. 79. 51. 221.5. 80. 52. 1.416. 81. 53. 2.257. 82. 54. 15.81. 83. 55. 0.2387. 84. 56. 0.03593. 85. 57. 7.212. 86. 58. 0.0001174. 87. 0.000000000004936. 12940. 99.82. 2.381. 2.155. 282.6. 1.706. 146.8. 1.938. 1.913. 1.616. 9.218. 2.468. 5.487. 1.523. 0.2356. 1.293. 0.7397. 0.02558. 1.568. 6.955. 0.002431. 0.3525. 0.0005218 0.01804. 33.04. 28.39. 12.58. 0.5096. ANSWERS. 49 Exercise CXIII. 1. 24:35. 4. 15:16; 23 :25 ; 10:11; 3:4. 6. 4 :15, 2. 10:9. 5. a + b-.a ~ b>. a' + b':a'-P. 7. 2 :3. 8. a* + a^x^ + x*:a*-x*. 10. 1 : 1. 13. 6, 8. 9. cc'-llx + 28:a^. 11. 18, 27. 14. 4 :1. 15. 1 : 5. 17 '^p-'^q 16. 26, 40, A to B. q-p 52, 60, B to 0. 18. 1240, 680 rods. Exercise CXIV. 16. 3, f lo b(a+by 19. ±9. ±3. 17. a-\-b, i{a~b). (a --by 20. 1. 4. 21. dz2. 25. 440 and 362 yards permin. 22. 8:7. 26. 45 and 30 miles per hour. 23. A's, $200;B's, $150. 27. 70 feet. 24. 6. 28. 3:7; 3/^ :7A. Exercise CXV. 1. 9|. 6. 64X'=9Y'. 10. Vol., 1611; 14. 87.1 dys. 2. f 7. X^=i^. dis., 402i. 15. 1 inch. 3. 15. ys 11. 9i. 16. 1 : 1. 4. 5. 8. 5. 12. 20^ feet. 17. 10. 5. S^. 9. 8. 13. 516|oz. 18. 18 feet. Exercise OXVI. 1. 53. 13. -29. a + 21b. -1. 0. 2. 4^. 3. 7i; 6;a^ 4. 5f 10, 14^; f 4i, Sf 12f 60 ALGEBRA. ^' ^^^ 7. 185. 35. ^(3n-l). -28. 3361. 6.-7. 2 8. 1, 2f, 3|, 5^, 6f , 8. 9. 9. 10. - 10. 11. 6, 14, 22. 12. 6. 15. 4, 15, 26, 18. 5, 7, 9. 13. 12. 16. 3, 5, 7; or 7, 5, 3. 19. - f, I, IJ. 14. 5. 17. 8. 20. 14. 21. 10 days. 22. 234. 23. 2, 6, 10, 14; or 14, 10, 6, 2. 25. 7 days. 27. 3,6, 15; or 15, 12, 3. 36. 6433i feet. 28. 10 seconds; HHe distance. Exercise CXVII. 1. 1458. 96. yfg. -128. x^\ 201 a' m\ 2. Qx'fVlbi. 3. 4. '4. 20, 50; 28, 56, 112. 5. 5th. 6. 3 ; 192. 7. 765. 547. 15M. 1953.1. ?5^. ^ 256 8. 10%. 9.3. 10, llfl. 12. i 1,2, 4. 13. ^V 14. 12, 18, 27. 16. 0.59049. 18. 1, 3, 9. 15. 1, 2i. 17. 2, 4, 6, 9. 19. 49, 1. 20. 1, 3, 9, 27; or 27, 9, 3, 1. 21. 139. 22. 8. H. i. ^. ^\. H. i If. il ||. Exercise CXVIIL' 1. 2|, 3, 4, 6. 4. 13-^, Uj\ 17|, 183%. 7. 8, 2. 2. 9,1; or f , -7^. 5. 6th. 9. 104, 234* 3. 12. 6. 6, 6|, 6f , 7y%. 10. 2, 3, 6. For answers to Exercises CXIX. and CXX. of Elements of Alge- bra, see answers to CXX. and CXXI. of this book ; and for answers to Exercise CXXI. pf Elements of Algebra, see Appendix- ANSWERS. 61 Exercise CXIX. 1. 454.053,600. 8. 1956. 15. 33. 2. 6. 9. 109,600 16. 720. 3. 6. 10. 2520. 17. 400,400. 4. 12. 11. 420. 18. 6160. 5. 4815; 969; 3876. 12. 5040. 19. 161,700. 6. 95,040. 13. 244. 20. 1,368,000. 7. 5040; 144. 14. 6720. 21. 15. \mx\n 23. 24. 1814400.25. 5775, (|rX|m — r)(|sX \n-s) 11; 3. 26. 36. ^ \^ . m 28. 80. 29. |9 |2. 30. 31. 325. 32. 7. '• [20 |70' [19 |70' 645. 33. 52. Exercise CXX. 1. 1 + 10a; + 40:^2 + 80a;3_|. 80 r?;* + 32a^. 2. o(^- 24:r^ + 252:r« - 1512:^;^ + 5670a;* - 13,608a;3 + 20,412a;2 _ 17496a; -f 6561. 3. l^x^-ma^y-\-2lQxhf-2mxif-{-^ly\ 4. S-\2x-\-Qa?-a?. . 15 y 45 y^ 135?/ 405 y* 243 .y^ 4 8 32 256 1024' 6. l_3. + 4.--— + — - — + — 7. - 14,080.000 rry. "" 243 "^ 729 ~ 19683' 52 ALGEBRA. g 85a:y 11. 70. ^^^^ 12. -252. S5x , 35y 10. -115,200 a; V- 7" ' "7~* 1X2X (r-1) ^^"^^ "^ • 1X2X (r-1) '^ ^ 16. H^-^) (^-^-^)a----3af-^3 1X2X (r + 3) 17. -±=a":r''. 18. 0. (^7 Exercise CXXI. ^2 8^16 2. l + lor-iar' + A^"- , ! f 1 , 3:7: 3a;' , 5a;' 3. a^\ 1+ + 4a 32a'' 128r/ j 4. l + 4x + 10a;' + 20:c' + If n be an integer, in the expansions of (l + a:)* or (1 — a?)-**, the coefficients are all positive; of (1 — .r)" or (1 + x)-^ they are alter- nately positive and negative. 5 5aV , 15 ax* 5x^ , 5. ,5___ + _^^.____ + 6. X-' - f :r-V + \^^~'y' - H^"^?/' + 7. (2a;)-i;i + |:r-V + AV^''2/'+TVA^~'y' + 1- 8. l-x-2x''~e>2^- ANSWERS. 53 * 8a4 "^Sa 128^2 1024 a^ )' 10. l + |y + ^Y + .%5^ + 11. l + |.r + f:r2-it-r« + 12. l-|a; + ¥a;2-H^ + ^ _ 1X3X5 (2.-5) ^3^^., h'-i X 2-^ The factor (— l)*" is introduced because the rth term will be positive or negative according as r is even or odd, 1x2 16. 8.06224. 16. 1.01098. 17. 2.002224. la l + ^x-^x^-\^x^+ 19. 351. Exercise CXXII. 1. h, i I. 2. f. 3. -iJ^. 4. f f 5. 7 to 2 against the event. 6. -f . 7. h 10. I". 14. Their chances are i, i, ^. 8. ^-^. 12. ^\. 15. $40. 17. 34 to 43. 9. f 13. I 16. 2 to 3. 18. ^^, ^- i^, ^, ff. 19. ^At- 20. t\, 3-b. 23. 3-^. 24. ^2^, -1%. fi. Exercise CXXIII. 1. f. 7. rl^. 15. f. 2. y-. 8. 49 to 15 against. 16. ff . n 9. ylfriy. 17. T^. 4. 1 - (tt)'. 11. Tfir. t'A. j%- 54 ALGEBRA. 6. 5 to 3 against. 12. ^, ■^, j\. 19. {{ 7 to 1 against. 13. ^, ^, fff, |||. 20. 1 - (ff )^ 6. 6 to 1. 14. 7 to 2. 21. y^. 250 276 7750 24. ^. 26. 131 to 112. A5 ' A5 ' f^5 ■ 25 1-15M 07 3497x5^ 23. ^, ^, ^, -i^. ' 5« ■ ^'' 8« • 28. 1||. 29. As 16 : 12 : 9. 30. $0.37i 7 to 1. 31. He may expect a loss of $0.25. 32. $6.93, $12.86, $3.96. ' ' 33. +$2e, -lit. -$2i|; +$2|, +$f -$2A. Exercise CXXIV. 1. Between 48 and 49. 12. 4|%, nearly. (ii\ log ^- log ^ ^ logi^ 13. $24.50. 14. 4%. 15. $7298. 3. $.00356. 16. A little more than 9 yrs. 4. ^ X 4.310. 17. A little less than 17 yrs. 5. $445.20. 18. $968.45. 6. $20,255, nearly. 19. 51.37. 7. 5.358. 20. 4.708. 8. $34.30. 21. ^%. 9. 22. 22. 2,428,900. 10. 115, nearly. 23. $27,657. 11. 5%, nearly. - ANSWERS. 55 Exercise CXXV. 1 3 + 3 + -1 2-f 1+ ' 1 + -1 • '^' ' 1+ ^ 2 + i. 3 + i i. 9+i 1 1 i. The fifth convergents are .1| ; j^ ; f || ; ^J^ ; ^i- 1 2+ 1 5 + —^ 4-+ ^ 2 + ^ 6 + 1 1 2 + -1 4+-1 ^' 1 The third convergents are ^ ; f|-|- ; ff ; |f. 3. 4 + ^ 4 + — ^ 1 + -^ 1 + — ^ - • 1 2 + — J— 4+ ^ l + -i- ■ 2-f ^ 8 4- etc. 8 + etc. 56 ALGEBRA. l + -i-y 2 + -1- 10 + etc. ^14 + etc. 5. H- 8. ^. 6. fW 9. ¥; M; Hi- 10. 15 and 16, 61 and 65, 76 and 81, etc. 11. Vl5; i(Vl97-13); ^(Vl093-13). 12. Limit of error = .00012. 13. {, ^, ^, ■^, .. 14.|,f,if Jfi Exercise CXXVI. 2"*" 4 "^ 8 "^ 16 "^ * 2 4"^ 8 16'"^ • i"^16"^64 256 4. l-^-a;8+:?;«+ 7. 4:^+2:^2- 8^-22^*- ■ 5. \-\-2x-\-a?~^a?— 8. y-'if + f-y* + 6. 5-17a:+56:r2-185a;3_|_ 9^ y^2y^+bi/-\-14:y^-\-"' •^3 15 315^ ^ 2^3 4^ l + 2:r 3-7a; 12. l-^+a;^' i._3^^^2' 4 + r-5a,-^ ANSWERS. 57 ,3 _3_4-_l- 15 _-§ L_ • ^_|_4"^^_5 • 3(a;-5) 3(2:^7-1) 14. -^_ 6 la 4 ■ ^ a: + 3 :i; + 4 7(^7+2) 7(a7-5) 17. -J___^+A_. * a; — 1 a;^ + a;4- 1 18 3-1-,^+ 3 4a; 8(a:-2) 8(a: + 2) 4 4 71 19. ~ —^+ '^ 25a; ba^ 25(a; + 5) ^.11 + 2^^4 10 3(a; + l) (x-iy S(x + 2) 21 10 4a; + 7 * 3(a;+l) 3(a;2-a;+l) Exercise CXXVII. 1. 77323. 3. 631. 5. 2.425. 7. 68. 2. 11860. 4. 120. 6. 766.5. 8. 255. 9. 3n(n + l)(n + 3). 10. ■J-n(n+l)(4n-l). Exercise CXXVIII. 1. 11. 4. 460. 7. 4095. 10. 5525. 2. 11879. 5. 1540; 2870. 8. 1840. 11. 4970. 3. 5610. 6. 190. 9. 25. Exercise CXXIX. ^1/111 1 1 1 Y * 3V1 2^3 71 + 1 n + 2 n + 3y' 11 18* . U 1 3 1 2(n + 2)(n+3) 4(n+l)(n-f2)(7i+3)(n-H) ' 96 58 ALGEBRA. 3 lA 1 \. I ' 8^4 2(n + l){n+2)J' 32 ^ 5 3w + 5 . 5 6 (n + 2)(n + 3)' 6 5 1 1 .1 ' 4 2(7i + l)(7i + 2)' 4 Exercise CXXX. 1. 2r*-f 3r8+8r2+7r + 6. 3. 12. 5. 1111. 2. 2rH7r2 + 20r2 4-23r + 20. 4. 9. 7. 20.2444. 8. 2709316. Exercise OXXXI. 6. 3. 1. x=7, y = -2. ^2.(^ = 1^' y x = 4c, y = 2. \x = 2, y = = 2. 2. = 10. 3. 4. 5. 6. 7. 8. 9. x=l, y^-1. 13 1^=13, 2/ :. = ^, y=5. l^ = -3, , ^ = 8, y = 4. 14.|^ = ^'2/^ ^ -, U = -3, a: = 2, y=l. "...=-. "■{::!:;;: ^ = 29, y = 23. r^ = -12 .r = 2, y = -l. ^^•U = -3,' = 3. -f- -4. -9. y = y^ -13. -4. 14. -4. 10. x = ~l, y = -\. 17. .r = 2, y = ^¥- 11. x==\, y = \. 18. r. = 6, y- :9. Exercise CXXXIV. 1. a.'»-.^•2 -44:2; + 84 = 0. 2. x^ ~ CI? -Vix" ■\- x-\-\2^0. 3. .7;^ + 6a7^- 13:^3 -78a;2_|_ 35^ _l_ 216 = 0. 4. .^^ + 0.075 :i-2-0. 105 .^• + 0.01=0. 6. .i-«-ll:r2 + 40^-50 = 0. ANSWEES. 59 Exercise CXXXV. 1. x^-Sx'-10x-\-24: = 0. 4. :?;3_i8a;2^108a;-216=-0. 3. a;^-4:x = 0. 5. x'-5x' + 4: = 0. 6. 4a;*-17ar^ + 4=0. . Exercise OXXXVL 1. -3, -8. 6. i(-3±Vl7), 1, f 2. -6,-17. 7. f5, ^b. 3. ±a, ±aV3. 8. ±2, 1±V^, — 1± 4. -2, i±v^, =bV^. 9. ^, ±v=n:. 5. I, -i. 10. I, |(-lq=V=^), 0. Exercise CXXXVII. 1. 4,K-7±V37). 6. -4,i(7±^ 2. -2, 3±V5. 7. 2,-3,-8. 3. 3, 2±V^. a 2, ±5. 4. 3, 3, 2. 9. 1, 9, -7. 5. -5, i-(5zfcV29). 10. -3, 7, -4. Exercise CXXXVIII. T53). 1. 1, 2, 3, -4. 2. 2,-1, 2±2V2. 3. 2, 4, -1±V^. 4. 3±V2, 3±V'2. 5. 1, 1, 2, -4. 1. ^=3. 6. 4, -2, -1± 7. 2, 3, IrbV^. 8. 2,-2, -l=bV6. 9. 13,-1,-1,-7. 10. 2±:V=nLl, ^(7=bV37). Exercise CXL. 1. Last derivative = 2. 3. Last derivative = 6. 2. Last derivative = 24. 4, Last derivative = 120. 5. Last derivative = 24. 60 ALGEBRA. Exercise OXLI. 1. 1, 1, 6. 3. -1, -1, -1. 5. 3, 3, 3, -2. 2. 2, 2,3. 4. 3, 3,-4. 6. 2, 2, 2,-6. 7. 5, 5, 1, -1. 9. 1. 1, 1. -^(1=fV=II). 8. 5, 5, 5, -2, -2. 10. 1, 1, -4, -4. Exercise CXLII. 1. a:3 + 2ar'-18a; + 18 = 0. 2. a^^+lOar'- 42a: -576 = 0. Z. x^-^ci?- 1350 a:* - 18,000 a: + 162,000 =x 0. 4. a:" + 3a:* + 24a:3-72ar'+ 1296a: -23,328 = 0. 5. a:*-8a:2H-4a:-224 = 0. Exercise CXLIII. 1. 3/^-83/'+ 12y + 9 = 0. 3. 3/« + 4y^-15y-18 = 0. 2. y*+2y»-82/'-6y-|-15 = 0. 4. y» - 5y = 0. 5. y*+17y« + 92y2 + 172y + 96 = 0. 6. y* + 4.8y3-17.86y2_56.688y + 90.4761 = 0. 7. y*-f 1.2y« + 1.54y2- 2.292y + 3.1981 = 0. 8. /+3.5y* + 4.50y3 + 5.75y^+1.8125y- 16.15625 = 0. 9. /-8y* + 20y-19y2_2y + 20 = 0. 10. y« + 0.2/4-1.60y*-1.640y« + 2.6240y2 - 3.70208y + 5.138944 = 0. Exercise CXLV. 1. 637. 3. 0. 5. 5225. 7. 347. 2. -596. 4. -499. 6. 82. 8. 60146, 9. -182. 10. 11659. ANSWERS. 61 Exercise CXLVI. 1. 1.+, 0.4+, -1.+. 2. 2.+, -0.6+, -1.+. 3. -1.+, 1.+, 4.+. 4. 1.+, 1.+, -3.+. 5. -7.+. 3.+. 1.+. 6. 1.+, 5.+, -0.6+. 7. -1.+, -3.+, -4.+. 8. 1.+, 6.+, 7.+. 9. 0.7+, 2.+, -0.7+, 5.+. 10. 2.+, 0.4+, 0.6+, -3.+. 1. 2.833066+. 2. 4.264429+. 3. 1.157451+. 4. 2.645751+. 5. -3.907378+. Exercise CXLVII. 6. 8.414455+. 7. 4.898979+. 8. 32.865384+. 9. 624. 10. 29. 11. 43. 12. 742.826359+. 13. 7.103575+. 14. -4.532088+. 15. -0.732050+. Example in § 622 : 3.353, 3.415. Exercise OXLVIII. :1, i(-7±3V5). 2. i(V5-l±V-10-2V5), -i(V5+l=FV-10+2V5). 3. ± 1, i(3 + V=^rfcV-14=b6V^7), i (3 - V-^ ± V-14-6V^. 4. 2±V3, i(l±V^). 6. -1, 2zfcV3, |(1±V^). 5. 1, l,i(l± V=^T5). 7. 3±2V2, 2±V3. 8. -l.i(l m Vm'-2m-3). 9. -1, i(l+V5±:V2V5-10), i(l-Vb±-yj~2Vb-10). 10. 1, iV-l, J(-1±V=^). Exercise CXLIX. 1. 2.438+. 5. 0.951+. 9. 1.775. 2. 2.096+. 6. 1.900+. 10. 2.925. 3. 2.872+. 7. 0.569+. 11. 15 years. 4. 2.970+. 8. 0.801+. 12. 9 years. 13. 20 years. 14. 8 years. 15. 17 years. 62 ALGEBRA. Exercise GL. 1. -2, -5, -5. 5. -6, 3±5V^=^. 2. 10, i(ll±5V=^). 6. 8, -4±3\/^. 3. 2, 2±V==1. 7. 1, 1, -2. 4. 1, -2=fc3V=n:. 8. -11, ^(iiiV^^n^), Exercise CLI. 1. 1.1542, -0.57709 =b 1.9998 V^. 2. -0.41813, 0.20906 ± 2.6704 V^. 3. -3.2264, 1.6132 =b 0.89836 V^. 4. 2.4567, -1.2283 d= 0.72557 V^. 5. 1, 1.5616, -2.5616. APPENDIX. ExEECiSE CXXI. {Elements of Algebra). * 32 16 8 4 2 *^ ■ o 32a^5 2Qai'^5^ Sa^S* 15ae5« 135 a^S^ 243Z>io 243 27 3 8 128 1024 * . 3. 0}%''^ + 10 a"&i V + 40 ai26"a;8 + gO a"6V2 + 80 a"6 V« + 32 ai^SV". ^ 128 g^S^* l\2aWy 14 a^o^V 35ai"&V 105a"6V 2187 243 9 12 32 _ 561a}^bY , 1701 a}^Y _ 2187 aV 256 2048 16384 5. a^Va + la^x + 21a'^Va X x^ + 35aV ^ s^aVa X x* + 21 ax^ + 7VaXiC* + a;''. 6. 16 ¥ - 64 J3 V2& X m + 224 6%2 _ 224 b^VUx m? + 280 6^ '- 1126 V26 X m» + 56 bm^ - SVU x m^ + m\ ANSWERS. 63 7. 27c3 V3 c + 378 c^a + IdQcWS cxa-' + 2520 c^a^ + 1680c\/3 c X a* + 2016ca5 + 448V37x a^ + 128al It) A A - 6804 aVfa X y^ + 10,206 ay^ - 1 7,496 VJa X y' + 6561 3/8. 9. 64 e^ + 192 ae2\/^ + 240 a V + 160a3ev^+ 60a*e + 12a5\/i' + a«. 10 64 gg 64a5V2^ 160 a^ 2,%)a?xyPIx ^^-'^-'^ T2^ 81 27 27 3 + 16aa;2\/2x + 8ar». ^j 243 g^ 405a^Vj^ 1350^3; \bc?xyf\x \bax^ x^y/\x 1024 256 64 16 16 4 ' 12. 64g6- 576a5v^ + 2160a*2/ - \Z%)a^yyfy + 4860ay -2916g2/2vV+729y3. 10 ^14 ^ 7g^^VI ^ 21 gio^ ^ 35 g^z VJ ^ 35 ah^ , 21 g^z^ yfl 16 32 TgV 2V2 64 128 ' 14. h" - 8 W ^hy + 28 h^y - 56 h'^yVb^ + 70 &y - 56 Jy^ v^ + 28by^-8y^Vby + y\ 15. 8c3+ 24 c2 Vec^^- 180 c2a;+ 120 ca^V6cx+270ca;2+54a;2\/6cx+27x3. 16. 8,200,192 g«. 18. 326,592 rc^^. 20. l^^'^^"^ 256 17 352,716 gii(f 10. 19. -12,033,222,8802/81. 21. -165a%2. 22. (1 + x)-^ ==l-x + x'^-x^+x*-a^ + l_l + i_i+l_l + = 0.9. 9 92 93 9* 95 23. {l+xf=l+^-^+^-^\^- = 1+1 -A 2 8 16 128 256 16 512 _27__^2^.+_^70L._ ^1.1726. 8192 524108 8388608 24 (l+a;)-^=l-^ + ?JL'_5£! + 35a^*_.63x5 = 0.998503367... 2 8 16 128 256 64 ALGEBRA. 25. 0-5 - 5a-«a; + 15a-7a;2 - 35a-8a;3 + 26. a-« + 6a-7a; + 21a-8x2 + 56a-»a;3 ^ 27. 1 +^ + ^1^+1^+ 64 6« 6467 25668 64 6* 28. 512 92122 92160^2 67584023 c9 cw cii ci2 ... 29. 1 9a; 45a;2 165ar« a' 2aw 4 a" 8 a" 30. 1 5a;' 15a;« 35a;» a^® a" a^* a^* 31. 1 6.^^21^^ 56a^^ «' a^V^ «* a*VS 32. V 6 + + 2v/6 86V6 1662V6 33. ^Vj a; 3x-^ 7x^ 4:y/b^ 32hW 12862^ 34. .+ « -% ^' 2a; 8a;' 16a;* 35. • a 1 1 1 2a 8 a' 16a5 36. ^ a 4a' 68 a' 9 81 2187 ' 37. 1 1 I 3 5 aVa 2aVa 80^^^ 16oi0v'a 38. 1 a 3a2 5a' a; 2.'c' 8a;5 ' lQ^^ 39. -f-^ xi« 14a;i5 1 3A SA^ 9 81 * 8 256 2048 13 ;i' 65536 40. ^ 2d 12d^ 88 d' ^o 1 , ^' , 5a;* 5 25 125 "■ 27 81 1458 , 35a;6 ^ 39366 43. 7.2801098. 45. 2.8716219 47. 4.073062.... 44. 4.431047... 46. 3.0385047 48. 3.87827796...-