algebra de baldor ejercicios resueltos

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  • 1. EJERCICIO 1 1. Una deuda se expresa en sentido negativo. Luego inicalmente el estado econmico de Pedro es - 60 bs. Al recibir 320 bs. Aumenta su capital tantos bs. como ha recibido; entonces la operacin queda expresada como se indica 60 + 320 = + 260 bs.EJERCICIO 3 1. + 32 m ; 16 m 2. + 10 m ; 4 m 3. 50 85 = 35 m 4. 6 11= 66m2. 1.170 1. 515 = 345 sucres 3. 200 + 56 189 = + $ 675. -8 6 =-48 m 9 6 =+ 54 m4. 665 1.178 + 2 . 280 = + 437 soles 5. 20 15 + 40 75 = $ 30 6. 67 + 72 16 + 2 = $ 9 7. 200 78 81 93 + 41 59 = 70 colones 8. 45 66 79 + 200 10 = 0EJERCICIO 2 1. 12 15 = 3!400 2 = + 800 m corredor 400 3 = 1. 200 m yo7. Los 40 pies de longitud del poste se reparten asi: 15 pies que sobresalen - se asumen en sentido positivo- . 25 pies que se encuentran enterrados. - se asumen en sentido negativo - . Al introducir 3 pies ms, se adicionan a los que estan bajo el suelo y se deben descontar de los que estan por encima. Aritmticamente significa:40 = 15 + 25 25 3 = 28 pies + 15 3 = + 12 pies2. 3 + 8 6 = 1! 3.6.15 ( 3) = 15 + 3 = 18!( )4. 8 + 5 = 8 + 5 = 13! 8. 55 52 = + 3 m5. 4 + 7 + 2 11= 6! 6. 8 + 4 1 = 8 + 4 = 4! 7 am 8 + 4 2 = 8 + 8 = 0! 8 am 8 + 4 5 = 8 + 20 = + 12! 11am9. 32 + 15 = 17m 10. 35 47 = 12 m 11. 39 + 56 = + 17m 12. 90 58 36 = 4 m7. 1 2 2 = 1 4 = 5! 10 am 1 2 3 = 1 6 = 7! 11am 1 2 3 + 3 1= 1 6 + 3 = 4! 12 am 1 2 3 + 3 3 = 1 6 + 9 = + 2! 2 pm 8. 56 + 7 = 49! 9. 71 + 5 = 66! long 15 + ( 5) = 20! lat.10. 18 + 3 =+ 21! long65 -4 =+ 61! lat. 11. -75 +135 =+ 60 aos13. 72 72 72 72 30 1= 72 30 = + 42 m 30 2 = 72 60 = + 12 m 30 3 = 72 90 = 18 m 30 4 = 72 120 = 48 m14. 120 ( 60) 1 = 120 + 60 = 60 Km 120 ( 60) 2 = 120 + 120 = 0 120 ( 60) 3 = 120 + 180 = + 60 Km 120 ( 60) 4 = 120 + 240 = + 120 Km

2. EJERCICIO 7. 1. x + 2x = 3x27. 11a + 8a + 9a + 11a = 39a 28. mx + 1 + 3mx + 1 + 4mx + 1 + 6mx + 1 = 14mx + 12. 8a + 9a = 17a29. x2 y 8x2 y 9x2 y 20x2 y = 38x 2 y3. 11b + 9b = 20b30. 3am 5am 6am 9am = 23am4. b 5b = 6b 5. 8m m = 9m31.6. 9m 7m = 16m1 1 1 4 + 2 +1 + 8 15 a+ a+ a+a = a= a 2 4 8 8 8 2 1 1 1 8 + 10 + 2 + 1 21 ax + ax + ax + ax = ax = ax 5 2 10 20 20 207. 4a x + 5a x = 9a x32.8. 6a x + 1 + 8a x + 1 = 14a x + 133. 0,5m + 0,6m + 0,7m + 0,8m = 2,6m9. mx + 1 5mx + 1 = 6mx + 134. 1 1 1 ab ab ab ab 7 14 28 4 2 1 28 35 5 = ab = ab = ab 28 28 410. 3a x 2 a x 2 = 4a x 2 11.1 1 1+ 1 2 a+ a= a = a =a 2 2 2 23 1 6ab + ab 7 ab + ab = = ab 12. 5 10 10 10 13.1 1 2 +1 3 1 xy + xy = xy = xy = xy 3 6 6 6 214. 1 4 1 4 5 xy xy = = xy = xy 5 5 5 515. 5 2 1 20 3 2 23 2 a b a 2b = a b = ab 6 8 24 247 87 15 16. a 8 a = 8 a = 8 a17. 8a + 9a + 6a = 23a2 3 1 1 1 3 x y x 3y x3y x y 3 6 9 12 2 4 6 4 3 3 37 3 x y= x y = 36 3635. 36. ab 2 + ab 2 + 7ab 2 + 9ab 2 + 21ab 2 = 39ab 2 37. m m 8m 7m 3m = 20m 38. x a + 1 8x a + 1 4x a + 1 5x a + 1 x a + 1 = 19x a + 1 39.1 1 1 1 1 a+ a+ a+ a+ a 2 3 4 5 6 30 + 20 + 15 + 12 + 10 87 29 a= a= a = 60 60 20 1 1 1 1 1 ab ab ab ab ab 3 6 2 12 9 12 6 18 3 4 43 = ab = ab 36 3640. 18. 15x + 20x + x = 36x 19. 7m 8m 9m = 24m 20. a 2b a 2b 3a 2b = 5a 2b 21. a x + 3 a x + 8a x = 12a x 22. 5a x + 1 3a x + 1 5a x + 1 = 13a x + 1EJERCICIO 81 2 6+3+4 13 a= a 23. a + a + a = 2 3 6 61. 8a 6a = 2a24. x 25.2 1 6 4 1 11 x x= x= x 3 6 6 61 3 2 + 3 + 10 15 3 ax + ax + ax = ax = ax = ax 5 10 10 10 2 3 45 62 2 2 26. a x a x a x =9 10 12 2 31 2 a x= a x 12 122. 6a 8a = 2a 3. 9ab 15ab = 6ab 4. 15ab 9ab = 6ab 5. 2a 2a = 0 6. 7b + 7b = 0 7. 14xy + 32xy = 18xy 3. 8. 25x 2 y + 32x 2 y = 7x 2 y35.9. 40x 3 y 51x 3 y = 11x 3y 10. m2n + 6m2n = 5m2n 11. 15xy + 40xy = 25xy 12. 55a 3b 2 81a 3b 2 = 26a 3b 25 m + 1 7 m + 1 10 7 m +1 3 m + 1 1 m + 1 a a = a = a = a 6 12 12 12 42 36. 4a 1 2 12 1 2 11 2 a = a = a 3 3 337. 5mn +3 20 + 3 17 mn = mn = mn 4 4 413. x 2 y + x 2 y = 038. 8a x + 2b x + 3 25a x + 2b x + 3 = 17a x + 2b x + 314. 9ab 2 + 9ab2 = 039. 15. 7x2 y 7x2 y = 0 16. 101mn + 118mn = 17mn7 m n m n 7+8 m n 1 m n a b +a b = a b = a b 8 8 840. 0,85mxy 0,5mxy = 0,35mxy17. 502ab 405ab = 97ab 18. 1024 x + 1018x = 6x 19. 15ab + 15ab = 0EJERCICIO 920.1 3 23 1 a a= a= a 2 4 4 41. 9a 3a + 5a = 11a21.3 1 32 1 a a= a= a 4 2 4 43. 12mn 23mn 5mn = 16mn22.5 2 5 2 10 5 2 5 2 a b a b= a b= a b 6 12 12 125. 19m 10m + 6m = 15m4 2 9 2 8+ 9 2 1 2 x y= x y= x y 23. x y + 7 14 14 14 3 5 3 10 7 am am = am = am 24. 8 4 8 8 25. am + 26.3 5+3 2 am = am = am 5 5 55 7 20 21 1 mn mn = mn = mn 6 8 24 242 27. a b + 11+ 3 2 3 2 8 a b= a b = a 2b 11 11 1128. 3,4a 4b3 5,6a 4b 3 = 2,2a 4b 3, 29. 12yz + 3,4yz = 2,2yz 30. 4a x 2a x = 2a x 31. 8a x + 1 + 8a x + 1 = 0 32. 25ma 1 32ma 1 = 7ma 1 33. x a + 1 + x a + 1 = 0 34. 1 m 2 1 m 2 1+ 2 m 2 1 m 2 a a + a = = a 4 2 4 42. 8x + 9x x = 0 4. x + 19x 18x = 0 6. 11ab 15ab + 26ab = 0 7. 5a X + 9a X 35a X = 31a X 8. 24a X + 2 15a X + 2 + 39a X + 2 = 0 9.2 1 2 + 1 3 0 y+ yy= y= y=0 3 3 3 310. 11.3 1 1 12 + 5 10 17 m+ m m = m = m 5 4 2 20 203 2 1 3+2 8 2 3 a b + a 2b a 2b = a b = a 2b 8 4 8 812. a + 8a + 9a 15a = a 13. 7ab 11ab + 20ab 31ab = 15ab 14. 25x 2 50x 2 + 11x 2 + 14x 2 = 0 15. xy 8xy 19xy + 40xy = 12xy 16. 7ab + 21ab ab 80ab = 53ab 17. 25xy 2 + 11xy 2 + 60xy 2 82xy 2 = 36xy 2 18. 72ax + 87ax 101ax + 243ax = 157ax 19. 82bx 71bx 53bx + 206bx = 0 20. 105a 3 464a 3 + 58a 3 + 301a 3 = 0 4. 21.1 1 1 1 30 20 + 15 12 13 x x+ x x= x= x 2 3 4 5 60 60EJERCICIO 1022.1 1 1 1 4 4 + 2 1 1 y y+ y y= y= y 3 3 6 12 12 121. 7a + 6a 9b 4b 9b 4b = 13b 7a + 6a = 13a = 13a 13b3 2 1 2 1 2 2 23. 5 a b 6 a b + 3 a b a b 18 5 + 10 30 2 7 2 = a b= a b 30 30 5 2 1 2 3 2 2 24. 6 ab 6 ab + ab 8 ab 20 4 + 24 9 2 9 3 ab = ab 2 = ab2 = 24 24 8 25. a + 8a 11a + 15a 75a = 64a 26. 7c + 21c + 14c 30c + 82c = 80c 27. mn + 14mn 31mn mn + 20mn = mn 28. a 2 y 7a 2 y 93a 2 y + 51a 2 y + 48a 2 y = 0 29. a + a a + a 3a + 6a = 3a 30.1 2 7 1 x+ x x+ xx 2 3 6 2 3+47+36 3 1 x= x= x = 6 6 23 1 5 31. 2x + 4 x + 4 x + x 6 x 48 + 18 + 6 + 24 20 20 5 = x= x= x 24 24 6 32. 7a x 30 a x 41a x 9a x + 73a x = 0 33. a x + 1 + 7a x + 1 11a x + 1 20a x + 1 + 26a x + 1 = a x + 1 34. a + 6a 20a + 150a 80a + 31a = 88a 35. 9b 11b 17b 81b b + 110b = 9b 36. a 2b + 15 a 2b + a 2 b 85 a 2b 131a 2b + 39 a 2b = 162 a 2b 37. 84m2 x 501m2 x 604m2 x 715m2 x + 231m2 x + 165m2 x = 1340m2 x38.5 3 2 2 3 2 1 3 2 5 3 2 a b + a b a b a b + 4a 3b 2 6 3 4 8 20 + 16 6 15 + 96 3 2 111 3 2 = ab = ab 24 24 37 3 2 = a b = 4 5 a 3b 2 8 839. 40a 81a + 130a + 41a 83a 91a + 16a = 28a 40. 21ab + 52ab 60ab + 84ab 31ab ab 23ab = 02. a + b c b c + 2c a aa =0 b b =0 =0 c c + 2c = 03. 5x 11y 9 + 20x 1 y 5x + 20x = 25x 11y y = 12y = 25x 12y 10 9 1 = 104. 6m + 8n + 5 m n 6m 11 6m m 6m = 13m 8n n = 7n = 13m + 7n 6 5. a + b + 2b 2c + 3a + 2c 3b a + 3a = 2a b + 2b 3b = 0 = 2a5 11= 6 2c + 2c = 06. 81x + 19y 30z + 6y + 80x + x 25y 81x + 80x + x = 0 19y + 6y 25y = 0 = 30z 30z7. 15a 2 6ab 8a 2 + 20 5ab 31+ a 2 ab 15a 2 8a 2 + a 2 = 8a 2 6ab 5ab ab = 12ab20 31= 11= 8a 12ab 11 28. 3a + 4b 6a + 81b 114b + 31a a b 3a 6a + 31a a = 21a 4b + 81b 114b b = 30b = 21a 30b 3 4 2 3 4 2 3 3 9. 71a b 84a b + 50a b + 84a b 45a b + 18a b 71a 3b + 50a 3b 45a 3b + 18a 3b = 48a 3b 84a 4b 2 + 84a 4b2 = 0 = 48a 3b 10. a + b c + 8 + 2a + 2b 19 2c 3a 3 3b + 3c a + 2a 3a = 2a b + 2b 3b = 0 c 2c + 3c = 0 8 19 3 = 14 = 2a 14 5. 2 2 3 2 2 3 11. m + 71mn 14m 65mn + m m 115m + 6m 3 3 3 2 2 2 m + 6m = 7m m 14m m 115m2 = 129m2mn 71 65mn = 6mn= 7m3 129m2 + 6mn 4 3 2 2 4 2 3 2 3 2 4 3 12. x y x y + x y 8x y x y 10 + x y 7x y 9 + 21x y y + 50 x 3 y 2 + x 3 y 2 7x 3 y 2 = 7x 3y 2 x 4 y 8x 4 y + 21x 4 y = 14x 4 yx 2y x2y = 0 y3 10 9 + 50 = 31= 14x y 7x y y + 31 43 2313. 5a x + 1 3b x + 2 8c x + 3 5a x + 1 50 + 4b x + 2 65 b x + 2 + 90 + c x + 3 + 7c x + 3 8c x + 3 + c x + 3 + 7c x + 3 = 0 = 25 3b x + 2 + 4b x + 2 b x + 2 = 05a x + 1 5a x + 1 = 0 50 65 + 90 = 2514. am + 2 xm + 3 5 + 8 3am + 2 + 5xm + 3 6 + am + 2 5xm + 3 x m + 3 + 5 x m + 3 5x m + 3 = x m + 3 =xm+3am+2am + 2 3am + 2 + am + 2 = am + 2315. 0,3a + 0,4b + 0,5c 0,6a 0,7b 0,9c + 3a 3b 3c 0,3a 0,6a + 3a = 2,7a 0,4b 0,7b 3b = 3,3b = 2,7a 3,3b 3,4c 16.17.5+86= 31 1 3 1 3 1 a + b + 2a 3b a b + 4 6 4 2 2 3 1 3 2+83 7 a + 2a a = a= a 2 4 4 4 7 17 1 = a b+ 4 6 40,5c 0,9c 3c = 3,4c1 1 2 18 1 17 b 3b b = b= b 3 6 6 63 2 1 1 m 2mn + m2 mn + 2mn 2m2 5 10 3 3 2 1 6 + 1 20 2 13 m + m2 2m2 = m = m2 5 10 10 10 13 2 1 = m mn 10 3 2mn 3 1 32 1 = = 4 2 4 4 6 1+ 6 1 1 mn + 2mn = mn = mn 3 3 318. 3 a 2 + 1 ab 5 b 2 + 2 1 a 2 3 ab + 1 b 2 1 b 2 2ab 3 4 2 6 4 6 3 238 9 3 2 7 2 9 + 28 2 19 2 1 3 a = a ab ab 2ab = ab = ab a + a = 4 4 4 3 12 12 2 4 5 2 1 2 1 2 5 + 1 2 2 6 2 2 b = b =b b + b b = 6 6 3 6 6 19 2 9 2 a ab b = 12 4 6. 3 2 2 1 xy 0,6y 3 x 2 y 0,2xy 2 + y 3 6 8 5 4 2 22 2 x y=0 0,4x 2 y x 2 y = 5 5 , 1 14 3 2,4 + 1 3 0,6y 3 + y 3 = y = y = 0,35y 3 4 4 4 = 0,175xy 2 0,35y 3 + 252 19. 0,4x y + 31+37 m 2 3 m1 1 m 2 1 + a 0,2a m 1 + b m 2 b b 50 5 25 5 3 m 1 3 m 1 3 + 15 5 m 1 13 m 1 a + a 0,2a m 1 = a = a 25 5 25 25 13 m 1 1 m 2 = a + b 25 50a 20. 25m1, , 3 2 3 16 2 14 2 xy 0,2xy 2 = xy = xy = 0,175xy 2 8 8 8 31 6 = 257 m 2 1 m 2 1 m 2 7 2 + 10 m 2 1 m 2 + b = = b b b b 50 25 5 50 50EJERCICIO 11 Para resolver los problemas del 1 al 18 las literales toman los siguientes valores: a = 1 b = 2 c = 3 m = 1/2 n = 1/3 p = 1/4 1. 3ab = 3 1 2 = 6 2. 5a 2b3 c = 5 12 23 3 = 5 8 3 = 1204 2 2 2 1 1 3. b mn = 2 = = 2 3 6 312.4a 4 1 4 2 = = = 3bc 3 2 3 18 9 2 1 1 5 2 2 20 2 5b m 4 = 5 = 60 13. = = 1 1 1 1 np 3 4 12 12 22 1 1 2 334. 24m n p = 24 231 1 1 6 1 = 6 = = 4 27 4 108 18 35.2 4 2 3 2 4 2 1 2 1 8 1 a b m = 1 2 = 4 = = 2 3 3 3 8 24 36.7 3 2 7 1 1 189 1 1 189 63 = = c p m = 33 = 4 2 12 16 2 384 128 12 1214.22 1 1 2 33b c a 7. m n p = 8. 9.1 1 1 1 1 = = 4 4 27 4 4325 b 1 c 2 5 2 1 1 a m = 1 2 6 63 25 1 5 = = 6 2 122bc 2 = 2 2 32 = 4 9 = 36 = 63 2 10. 4m 12bc = 4 4 3 11. mn 8a b =15.1 3 12 2 32 = 2 3 216 = 2 6 = 12 21 1 1 8 4 8 14 23 = 64 = = 2 3 6 6 316.23 3 3 3 3 8 24 b 2 6 4 =4 = 4 = 4 = =1 2 2 2 2 2 9 18 6 c 3 3 3 3 31 2 = 1 = 1=3 = 2 1 1 n2 1 9 3 32m21 1 24 2 3 = = 2 2 2 n2 p2 1 1 2 2 3 4 24mn=4 4 = = 24 1 1 2 12 624 4 6 = 1 1 1 2 9 16 144 7. 3 3 6 3 3 64 2 3 36 17. 3 64b c = = 3 3 64 8 729 = 3 3 373.248 = 3 72 = 216 1 2m 2 218.3 1 3 3 1 2 2 apb 2 5 4 5 5 = = 3 3 125bm 3 3 125 2 1 3 3 2 2 2 24 3 3 1 4 5 5 = = = 250 3 3 125 3 5 2 2 23 5 = 6 = 2 15 75 25 2EJERCICIO 12 Para resolver los problemas del 1 al 18 las literales toman los siguientes valores:a = 3 b = 4 c = 1/3 d = 1/2 m = 6 n = 1/4 1. a 2 2ab + b 2 = 32 2 3 4 + 42 = 9 24 + 16 = 1 1 321 1 1 3 2 222 2 2. c + 2cd + d = + 2 + =3.1 1 1 4 + 12 + 9 25 + + = = 9 3 4 36 36a b 3 4 + = + = 9 + 8 = 17 1 c d 1 3 21 c m 6 2 2 72 + 6 64 +2= 3 + 2 = 24 + 2 = = 4. 1 1 3 3 d n 3 2 4 5.a 2 b 2 m2 32 42 62 9 16 36 + = + = + = 3 8 + 6 =1 3 2 6 3 2 6 3 2 66.3 1 3 1 1 1 1 1 10 + 5 4 c b + 2d = 4 + 2 = 2 + 1= = 5 2 5 3 2 2 5 5 51 1 3 4 ab ac bd 3 4 3 2 = 12 + 1 2 = 48 + 2 1 = 144 + 6 1 = 149 = 49 2 + = + 3 1 1 6 1 1 7. n 3 3 3 6 d m 4 2 2 4 8.b + n + 6m = 4 +1 1 4 + 1+ 12 17 1 + 66 = 2 + + 6 = = =8 2 4 2 2 2 1 31 21 42 2 9. c 3a d 16b + n 8d = 3 3 16 4 + 8 10.ma 63 216 = = 216 16 = 3.456 4 = b 1 d 1 16 21 1 1 1 1 2 16 + 1 13 = 3 16 + 2 = 1 8 + = = =6 2 3 2 4 2 2 21 2 8. 22 1 1 1 1 3 4 11. 3 c 2 + 4n = 3 + 4 = 3 + 4 = 1 + 1 = 2 + 1 = 3 = 1 4 4 6 4 6 12 24 24 24 8 m 22212.4d 2 16n 2 2 8 1 1 1 1 + 1= 2 + 8 1 = + 1 = + 1= 1 1 = 0 2 4 2 2 4 16 2 213.a + b b + m 3 + 4 4 + 6 7 10 = = = 21 20 = 1 1 1 1 1 c d 3 2 2 314.1 2 ba mb 43 64 + + 5a = + + 5 3 = + + 15 = 4 + 4 + 15 = 23 1 1 1 1 n d 4 2 2 412c a 16n a 1 + = 15. m d 2b1 1 3 16 3 1 1 3 4 + 48 47 1 43 43 3 4 + = +2= +2= = = 1 23 24 1 8 6 8 6 24 24 24 6 212 3 3 66 3a 6m 9 = 44 + = 16 + 3 6 3 6 316.4b +17.b + 2d 3c + 8d = 2 42 a b 3 2 + d 18. + 3 4 2 224 + 2 21 23 36 3 6 = 4 + = 4 + 1 1= 4 6 3 61 1 + 8 3 2 2 + 1 1+ 2 3 3 6 3 3 = = = = 4 2 4 2 4 4 4 1 3 2 + 229 3 2 3 4 1 2 144 1 4 a n = + 3 = + 3 3 4 4 3 4 2 3 9 2 12 3 2 3 24 2 3 9 3 64 + 9 12 61 5 = + = + = 8+ = = =78 3 4 2 3 4 2 8 2 8 8 22EJERCICIO 13 Para los problemas 1 al 24 las literales toman los siguientes valores: a = 1 b = 2 c = 3 d = 4 m = 1/2 n = 2/3 p = 1/4 x = 0 1. (a + b) c d = (1 + 2) 3 4 = 3 3 4 = 9 4 = 5()() (()()())2. a + b b a = 1 + 2 2 1 = 3 1 = 31 4 1 9 2 23 777+82 2 + 4= + 4= + 4= 3. b m c n + 4a = 2 3 + 4 1 = 2 3 2 3 2 3 2 24.1 1 (2m + 3n)(4 p + b ) = 2 2 + 3 2 4 4 + 2 = (1+ 2)(1+ 4) = 3 5 = 15 3 2()()(2)11()2()2 2 2 2 5. 4m + 8 p a + b 6n d = 4 + 8 1 + 2 6 4 = 2 + 2 3 0 = 0 2 3 4=15 1 =72 2 9. ()()()() ()(1 ) 2)( 2 1 1 = 4 41 46. c b d c b a m p = 3 2 4 3 2 1 = 1 1 1 ()()(7 168 7 161 1 2 3+ 4 + + 2 0 = 4 7 = = 26 5 = 28 = 6 6 2 3 6 6 6)2 2 2 2 7. b c + d a m + n + 2 x = 2 3 + 4 1 ()(1 2)()2 2 2 2 2 2 8. 2mx + 6 b + c 4d = 2 0 + 6 2 + 3 4 4 = 6 4 + 9 4 16 = 6 13 64 = 78 64 = 141 1 16 8 8m 16 p 2+ 4 1 = 4 + 4 = 4 + 2 = 2 + 2 = 2 + 6 = 8 = 2 2 + a= 3 9. 9n 18 2 6 b 2 3 3 3 9 2 3 3()b c a 10. x + m a + d c = 0 +()1 2 3 1 1 1 (1 + 4 3 ) = 2 (1+ 64 3) = 2 62 = 31 2 1 1 4 + 2 2 3 5 3 27 4 1 + 2 2 + 1 1 + 4 = 4 = 4 = = =52 5 2 4 1 3 9 4 9 5 5 94 m + p a 2 + b2 2 11. = 2 ac()()(12. 2m + 3n + 4 p 8 p + 6n 4m 9n + 20 p)1 1 2 1 2 1 2 1 = 2 + 3 + 4 8 + 6 4 9 + 20 = ( 1 + 2 + 1)(2 + 4 2)(6 + 5) = 4 4 11 = 176 2 2 3 4 3 4 4 3()()(2 2 2 13. c m + n d m + p + b n + p1 = 32 + 2)2 7 3 11 21 44 126 144 + 44 26 1 1 2 1 2 4 2 + + 2 2 + = 9 16 + 4 = 12 + = = = 2 12 = 2 1 6 2 4 3 4 3 6 4 12 2 12 12 12 c2 + d 2 32 + 4 2 2 1 9 + 16 2 1 25 2 1 2 1 5 1 m= = 14. 2 1 2 2 = 1 2 2 = 5 2 = 2 = 2 2 1 a 4 d 15. (4 p + 2b)(18n 24 p) + 2 (8m + 2)(40 p + a ) 2 1 1 1 1 = 4 + 2 2 18 24 + 2 8 + 2 40 + 1 = (1 + 4)(12 6) + 2 (4 + 2) (11) = 5 6 + 2 66 = 30 + 132 = 162 4 2 3 4 4 2 2 2 5+ 1 d 2 4 2+ 4 1 6 a+ 5 + 2 1+ 2 b m = 2 2 4 = 2 5 + 8 = 3 5 + 8 = 3 13 = 3 208 = 624 = 312 = 16. 1 d b p2 4 2 1 2 2 2 1 2 1 2 1 2 2 16 16 16 16 4 5+()( )1 2 2 321 2 2 32 62 90 2 88 = = 14 4 = 14 2 6 3 6 6 62 2 2 17. a + b c + 8b m n = 1 + 2 3 + 8 2 = 3 25 = 3 5 = 15 = 2 1+ 3 6 a+c 6n 1 4 4 1 2 4 3 (c + d ) p = 7 = = + + (3 + 4) = + 18. 2 2 2 4 2 2 2 7 7 b 2 10. ()()2 n 6 1 1 2 = 3 (3 2) 32 2 (4 1) 16 = 3 16 6 4 = 3 4 6 2 3 = 12 12 3 = 3 2 2 4 2 319. 3 c b 32m 2 d a 16 p 20.6abc 3mn cdnp + 2 8b 2(b a ) abc 1 2 2 1 3 3 4 2 3 3 4 = 36 + 1 2 = 6 + 1 1 = 18 + 12 8 = 22 = 11 = + 1 2 3 24 24 12 2 8 2 2 16 2 6 8 2 3 2(2 1) 6 1 2 321.a 2 + b2 12 + 2 2 5 5 5 + 216 221 + 3 (1 + 2)(2 1 + 3 2) = + 3 3 8 = + 72 = = = 73 2 3 2 2 + 3 (a + b)(2a + 3b) = 2 b a 2 12 3 3 3 3 1 1 1 1 1 1 + + + + a b b c n m22 22. b + 2 2 2 5 7 21 49 70 35 3 5 3 1 1 1 1 1 1 2 1 = = = 17 2 = 2 + + + + + = 4 + + + 2 = 4 + + = + 2 6 2 1 2 2 3 2 1 4 2 4 4 4 2 3 2(2m + 3n)(4 p + 2c) 4m 23.2n22 1 1 4 4 4 189 4 185 1 1 2 5 = 2 + 3 4 + 2 3 4 = (1 + 2)(1 + 6 ) 4 = 3 7 = 21 = = = 20 9 2 2 3 3 4 4 9 9 9 9 9 22c 2 2 3 22 3 3 = 3 3 = 3 4 = 6 4 = 54 28 = 26 3 n = 1 3 7 9 7 9 1 1 24. 2ab m b m 63 63 4 2 2 1 2 2 2 2 2 2 b2 EJERCICIO 14()10. $ x + a m1. a + b + m 2. m2 + b3 + x 4()11. m a b c Km3. a + 1 ; a + 212. $ (n 300)4. x 1 ; x 213.5. y + 2 ; y + 4 ; y + 614. $ 8a ; $15a ; $ ma()6. $ a + x + m 7. m n((365 x) das15. 2a + 3b +)17. 23n m2( ) ( 20. (a + b)( x + y ) 21. $ (8 x + 48) 22. bs. (a 8)(x + 4)19. $ 3a + 6b ; $ ax + bm23.75 sucres x16. a b m29. x m Kmc 218. x 2 m28. bs. x 6()24. $a m) 11. 3.000 25. n 1 colonesa +b 29. $ m 2x 26. a 3 soles27.m m 1428.30. x + 2 x + x +1 km h ax hab. 22a a. sucres 31. 1000 a + + 3 4 EJERCICIO 15 3 2 x y 8 32 + 3 2 3 29 2 x y= x y 4x 2 y + x 2 y = 8 8 81. m, n m + n20. 4x 2 y ,( )2. m , n m + n = m n 3. 3a , 4b 3a + 4b 4. 5b , 6a 6a + 5b( )5. 7 , 6 7 + 6 = 7 6 = 121.6. 6 , 9 6 + 9 = 3 7. 2x , 3y 2x + 3y 8. 5mn , m 5mn + ( m) = 5mn m 9. 5a , 7a 5a + 7a = 12a()3 3 mn , mn 8 4 3 3 36 3 mn mn = mn = mn 8 4 8 822. a , b , c a + b + c 23. a , b , c a b + c 24. a , b , 2c a b + 2c10. 8 x , 5x 8 x + 5x = 8 x 5x = 13x25. 3m , 2n , 4p 3m 2n + 4p11. 11m , 8m 11m + 8m = 3m26. a 2 , 7ab , 5b2 a 2 7ab 5b212. 9ab , 15ab 9ab 15ab = 6ab 13. xy , 9xy 9xy xy = 10xy()14. mn , 11mn mn + 11mn = mn 11 mn = 10mn1 2 1 2 a b 15. a , b 2 3 2 3 16.3 3 3 3 b, c b+ c 5 4 5 417.1 2 1 2 3 b, b b+ b= b=b 3 3 3 3 318. 1 1 1 1 2 xy , xy xy xy = xy = xy 2 2 2 2 23 2 abc , abc 5 5 3 2 5 abc abc = abc = abc 5 5 519. 27. x 2 , 3xy , 4y 2 x 2 3xy 4y 2 28. x3 , x2 y , 6 x3 x 2y + 6 29. 2a , b , 3a 2a + 3a b = 5a b 30. m , 8n , 4n m 8n + 4n = m 4n 31. 7a , 8a , b 7a + 8a b = a b 32.1 2 3 x, y, x 2 3 4 1 3 2 46 2 1 2 x x+ y= x+ y= x+ y 2 4 3 8 3 4 3 3233. 5 m , m , 3 mn 3 2 35 2 8 2 m m mn = m mn = m mn 5 3 5 3 5 334. 7a 2 , 5ab ,3b 2 , a2 7a 2 a 2 + 5ab + 3b 2 = 8a 2 + 5ab + 3b 2 12. 35. 7mn2 + 17mn2 5m 4m = 10mn2 9m 36. x 7x 8x y + 4x y + 5 = 6x 4x y + 5 33223EJERCICIO 16237. 5x2 x 2 + 9xy 6xy + 7y 2 = 4x2 + 3xy + 7y21. 3a + 2b c 2a + 3b + c5a + 5b38. 8a 2b a 2b + 5ab 2 11ab 2 7b 3 = 9a b 6ab 7b 2232.3 2 2 2 3 39. m 8m n + 7m n + 7mn n 3 2 2 3 = m m n + 7mn n40.41. a a 3b + 4b 8c + 8c = b3.m+np mn+ p4.42. m3 + 5m3 5m3 4m2n 4m2n 7mn29x 3y + 55.2 2 2 2 44. 6a + 9a 5ab 7b 8b 11a+ b c 2a + 2b 2c 3a b + 3c245. x y + x y 5xy + 7 xy 4y 8 2 2334= 2xy 3 4y 4 81 1 46. 3a a + b b 4 + 6 2 2 6 1 1 2 5 1 = a+ b+2= a b+2 2 2 2 2 47.1 2 3 2 2 1 5 5 x + x + xy xy + y 2 y 2 2 4 3 3 6 6 2 + 3 2 2 1 5 2 1 = x + xy = x + xy 4 3 4 348. 5ax 5ax 6a x + 1 + a x + 1 + 5a x + 1 + 8a x + 2 = 8a x + 23 2 2 2 1 1 49. x + x xy xy + y 2 + 5y 2 4 3 3 3 3+ 4 2 3 1+ 15 2 7 2 16 2 x xy + y = x xy + y = 4 3 3 4 3 50.3 2 1 1 1 5 a b a 2b + a 2b + ab 2 + ab 2 ab 2 4 4 2 2 6 3 1+ 4 2 3+35 3 2 1 2 = a b+ ab = a b + ab 2 4 6 2 6ay + az 12. 5x 7y + 8 4x y + 6 3x + 8y + 9 2x+ 2313. am + 6mn 4s am 5mn + 6s3am 5mn 2s am 4mn2b= 15a 5ab 15b 11 2 2 5 ax 7ay 6az 4 ax + 9ay + 8az3x= m3 8m2n 7mn22ax ay az11.0 x y +4 5x + 4 y 943. 9x x 11y 6y + 4z 6z = 8x 17y 2zab + bc + cd 8ab 3bc 3cd 5ab + 2bc + 2cd 2ab7a 4b + 5c 7a + 4b 6c c1 1 2 1 a a + b+ b6 2 4 3 5 2 1 10 + 3 1 13 = a+ b6= a + b6 4 15 4 1510.6.p+ q+ r 2p 6q + 3r p + 5q 8r 4r14. 2a + 3b 6b 4ca+ 8ca + 9b + 4c 15. 6m 3n7. 7x 4y + 6z 10x 20y 8z 5x + 24y + 2z 2x 4n + 5p 5p5m 7n 16. 2a + 3b8. 2m + 3n 6 3m 8n + 8 5m + n 10 4m 4n 89. 5a 2b 3c 7a 3b + 5c 8a + 5b 3c 6amc8a+ 5c 4 +6 + 7c 910a + 3b + 12c 7 17. 2x 3y6x+ 5z + 9 4 5 3y8x+ 5z 13. 18. 8a + 3b c 5a b + c 8a + 5a 9a xa b c 7a b + 4c19. 7x + 2ym24.n 11a x + 5am + 16an44m+ 9y 6z + 5 y + 3z 6 8 x 3y 5 7m+ 2bc + 2cd 3de 2ab + 4bc + 3de ab 3bc 6cdm2ab26. a b 23.bc c+d8x + y + z + u 3x 4y 2z + 3ucd am2 + n22aa 3 2a 2 4ma ab + b 2222a a 2 11a + 15 3mn28. 3x + x 33x 7x 2 x2 8x + 9x 6 2+ 2x3 3. x x2+4x x + 2x + 4 32x 2 3xy + y 29. x + 3xy 2y 2 3a 2 + a3x + 3xy 2y 2+ 4aa 4 + a 3 3a 2 + 4a10.+ 5a b 4ab 2a 2 7ab2 b33a 2 3ab + b 2+ xy 2 + y 315. x 3+ a 5ab b x 2 + 3x2 2a + 8ab b22x 324x 3 5x 2 y 3xy 2 5y 32+6bx 3 x 2 + 3x + 66. x 2 4 x 7x + 6 3x 222a 3 + 5a 2b 11ab 2 2b 32x3 b33 14. a2x 2 + 3xy y 2 4 4. a+ 8x 5313x+53 x + 4x 6 3 x2 + x 613. 4x 2 + 52+6+ a 7a + 4 5m 5n 22x 2 x5. 4a + 53 12. a+ 4n2 3mn1. x + 4 x x 2 5x+d day + 3z + 2u7.ca4x + 5y + 3z 4u 9x y + z + 2u22 2. a + ab 2ab + b 2+ 2c25. 5ab 3bc + 4cda+35ma + 1 11ma + 2 + 6ma + 3 m + 3n + 2p 9EJERCICIO 17a+2 5ma + 1 + 3ma + 2 + 12ma + 315x + 7y 3z 1020. m n p m + 2n 5 6m + 3p + 4 5m + 2n 8 3aa +1 a+2 5ma + 3 22. 6m 7m a +1a+ b c +d a b + c d 2a + 3b 2c + d 3a 3b + 4c d 14a x + 7am+ 3c19a5a x 3am 7an21.54x 2 11x + 111. 7x 2 + 5x 64 x + 8x 9 2x2 7x + 14 4 x 2 + 6x 1x 5x y 3 y32 4xy 2 5y 3 7m2n16.+m2 m + 7m n 3+ 4n3 + 6mn n33 2+ 5n3 + 6mn2 + 8n3 14. + x 3 4x 2+ a+5 4a + 4a m 3am2+ a4 +a 3a6 3a + 3a m 6am63+6+ a + a + a 2a a 543+ 3x 4 7x 2 3x+ 5xy 4 + 2y 52x 5 + 2 x 4 y + 2 x 3 y 2 + 3x 2 y 326.a 5 + a6 + a2 4x + 5+ a 4 + a3 + 6 + 3a 2 +a20. a + a + a2 + 5+ 6 5a+a +a +4643 b4a427.+ 4a + 7 a 2 8 + 5a 4a 253 a 3b + a 2b 2 ab 3 8 a2 6 3a + 5a 3b 4a 2b 2 41 4a 3b + 3a 2b 2 2a 4 x 2y2 5x 3 y 4xy y 5 4x5a4 21. x y5+ 3x y2x 5 + 3x 4 3x 3 7x 2 3x + 2a 3 + 5a+ 3x y2 33 2+63 xy 4+ 2x y+ x 95 19. x2 x 3y 2 4+ a 3 a 2 14 625 25. x+832+ 7a 2m 4am2x 4 + x 3 + 2x 2 3x + 115 5am2 + m33 3+ 7x 2 4x + 66 18. + a 8a 2m + 6am2 m324. x2 + x4 17. x+ 6xy 3 4x y ab 3 4b 4m3 n3 + 6m2n28. 4 xy 3 + y 4+ n3 4m2n + 5mn2 62 2 3b 4x 4 5x 3 y 5x 2 y 2 + 2xy 3 + y 4 6m n3+ 6mn23 2m + n 2m n 33211mn2 2 22. + x + xy+ 5a x 2 + 6xy + 5y 2x 1+ ax 1 7 x 2 + 7 xy y 2a + 6a xa 3 8ax 2 + x 3 3a + 14ax x 35a 32 13a x 3 3a+ ax 4ax+3+ 4ax3 2x 3x2+ ax + 2 3a2 x 3 5a 2 x3a 3x 130. a x + a x + 1 6ax x + 5a x 2+ 6a x 3 + 7a x 3 + a x 4 6x 2 4xy + y 223. 3a x 2x 29. a x 2 + 4 xy 7y 2ax 1+ax+3 5a x + 2 +ax 1a 2a x + a x + 1 + a x + 3 ax+3 3ax+2x2+ax2+ ax + 2 3a x + 2x +1 2ax 15. EJERCICIO 18 1.1 xy 3 1 xy + 25 2 x 6 1 x2 + 6 1 x2 + 36.1 2 x + 21 + y2 41 2 2+3 1 1 5 1 x + xy + y 2 = x 2 + xy + y 2 2 6 4 2 6 4 1 2 2. a + 2 ab 1 ab 4 1 ab 41 2 b 2 1 b2 52 xy 3 1 xy 6 5 xy 6+8.+ y22 xy + 5 1 + xy + 10 1 2 y 2 1 2 y 6 1 2 y 32 2 a 3 5 2 a 6 1 2 a 121 ab 5 1 ab 10 1 + ab 20 ++b32b 3+3 3 b 5 x2 + 5 2 3 x 3 5 3 x + 6+ 3 4 x 52 3 m 31 mn2 + 4 1 mn2 82 3 n 5 3 3 n 5 +n3m31 2 b 2 1 2 b 6 1 2 b 3 3 + 1 2 2 8 + 10 1 2 4 2 + 1 a + ab + b 12 20 6 17 2 3 4 17 2 3 2 = a + ab b2 = a + ab b2 12 20 6 12 20 35 2 a b 6 1 2 a b 23 x 8 3 x 453 4 2 4+5 3 36 x x + 2+ x + x 5 6 8 2 9 9 2 3 9 = x 4 x2 + 2 + x3 x x4 + x 3 x 2 x + 2 5 6 8 5 2 8 9. 3 + 1+ 2 2 3 2 3 3 2 4 +1 x + xy + y = x xy 4 10 6 4 105.x44 1 5 3+2 2 xy + y 6 3 2 5 1 5 = x 2 xy + y 2 = x 2 xy + y 2 6 3 3 3+3 +2 + y2 33 2 x 44 + 1 3 4 3 2 5 10 3 3 5 3 2 a + ab + b + a b 4 8 5 6 5 7 8 2 5 1 7 8 = a 3 ab 2 b 3 + a 2b a 3 + a 2b ab 2 b 3 4 8 5 6 4 3 8 5x2 +4.1 2 ab 2 3 2 ab 81 3 a 42 1 1 52 2 3 2 a + ab + b = a2 + b 4 10 10 22 3. x3 xy 4 1 xy 2 5 xy 6+5 1 2 2 16 + 3 + 6 2 9 6 + 10 x + y + xy 6 24 12 2 7 2 13 1 13 7 2 = x2 y + xy x 2 + xy y 6 24 12 3 12 24 7. a 3+2 2 y 3 1 2 y 8 1 2 y 41 2 m n 6 1 2 m n 2+2 + 3 3 2 + 1 2 2 3 5 3 1 3 2 m + mn + n + m n 3 8 5 6 5 1 6 2 5 1 1 6 = m3 mn2 n3 m2n m3 m2n mn2 n 3 3 8 5 6 3 3 8 5 4 10. x5 4 x 6+2x 2 y 2 + 3 2 2 x y 8 1 2 2 x y 4+2 4 y 7 1 4 y 14 1 4 + y 71 3 xy 6 5 3 x y 66 5 4 16 + 3 2 2 2 4 1+ 2 4 1 3 5 3 x + x y + y xy x y 6 8 14 6 6 1 17 2 2 5 4 1 3 5 3 = x4 + x y + y xy x y 6 8 14 6 6 1 4 5 3 17 2 2 1 3 5 4 x x y+ x y xy + y 6 6 8 6 14 16. 2 3 x 35 11. x + 3x 5 1 3 x 6 1 3 x 12+ 2x 5 +4 x 5 1 x 106 13. a +3 2 x 8 1 2 x 42 4 x 33 a2x 7 2 1 a3 + a2x 3 25 2 ax 6 7 2 ax 8 1 2 ax 45 2 a 8+67 3 4 85 2 3 5 3 3 43 a + a + a a + a 7 8 5 8 8 10 4 3 2 3 5 3 3 7 = a6 a + a + a a a 7 8 5 8 8 3 5 10 4 3 3 3 2 7 a6 + a a a + a a 5 7 8 8 8 a6 +7 3 13 1 2 x + x + x2 x4 4 12 10 8 3 2 7 3 1 2 13 2x 5 x 4 x + x + x 4 3 12 8 10+4 8 + 2 1 3 8 1+ 6 32 2 2 4 x + x+ x x 4 12 10 8 32 3 a 93 4 a 73 5 3 3 1 a a a 5 8 23 a 6 8= 2x 5 12.a2 +3 x 5++a41 3 x 3 1 3 x 9 14.x51 3 2 3 4 x y xy 10 4+ 2 3 2 x y 5y5 1 5 y 6 1 5 y 9 1 5 y 3+3 4 5 x y x2y3 5 6+2x 4 y1 4 3 2 3 4 18 3 2 6 5 3 + 10 4 5 x y xy + y + x y x2y3 10 4 18 5 6 3 3 2 3 4 29 5 13 4 5 x y xy y + x y x2 y 3 = x5 10 4 18 5 6 13 4 3 3 2 5 2 3 3 4 29 5 x y x y x y xy y x5 + 5 10 6 4 18 x5 +26 3 6+7 2 20 21 6 2 3 1 3 a + a x+ ax + x 9 14 24 9 4 1 2 7 4 = a3 + a x ax 2 x 3 9 14 24 9EJERCICIO 19 Para los problemas 1 al 14 las literales toman los siguientes valores: a = 2 b = 3 c = 10 x = 5 y = 4 m = 2/3 n = 1/5 1.3.4x 5y 3x + 6y 82x 4 4x x+ y= 54 + 7 5 3 4 5 52 4 2 + 10 5 4 3 4 4 8 = 625 + 28 125 5 25 16 + 50 64 256 8 = 625 + 3.500 2.000 + 3.200 256 8 = 3.811x 2 5x + 8 6x 2 + 5x 50 6x 2 + 10x 72 6 5 2 + 10 5 72 = 6 25 + 50 72 = 150 + 50 72 = 172+ 7x 3y + 10xy 3 x 4 + 7x 3 y 5x 2 y 2 + 10xy 3 y 4 8 24 8 = 8 8 = 0 x 2 + 10x 30 5x 2 y 2 84 x 4 y 4 5x 2 y 2 8 + 7x 3 y + 10xy 32y 82.x4 y44.3m 5n + 6 6m 20n + 8 12m 20n 12 9m 45n + 2 2 1 18 45 9 45 + 2 = + 2 = 6 9 + 2 = 1 3 5 3 5 17. 5.nx + cn ab 8nx 2cn ab10. ab 5nx 6 + x 4 x 2y25x 3 y10nx cn 3ab 5 1 1 10 5 10 3 2 3 5 5 5 50 10 = 6 3 5 5 5 = 10 2 18 5 = 15 6.x 3 y xy 3 + 5 6xy + 2 + 3xy 3 + 1 y46x 3y 4xy 3 + 2 + x 4 y4 x + 6x y 4xy y + 2 4334= 5 4 + 6 53 4 4 5 43 44 + 2 = 625 + 24 125 20 64 256 + 2 = 625 + 3.000 1280 256 + 2 = 2.091 .+ b3a3+ x2 y23 3a b + 8ab b 3 22 5a 3 6ab 2 + 3a b 4a+8 2b211.32 4 2 + 223 23 + 8 = 4 8 + 4 9 2 27 + 8 = 32 + 36 54 + 8 = 42 323+3 2 2 +1 6 + 1 3 2 a ab + b 4 6 9 3 1 4 = a 2 ab + b 2 4 6 9 3 1 4 22 2 3 + 32 4 6 9 3 1 4 = 4 6 + 9 = 3 1+ 4 = 6 4 6 9+ 125n 33 7. 27m 9m n + 25mn 22 14mn 28+ 10m n + 11mn 2 227m3 + m2 n + 22mn 2 + 125n 3 8 32232 1 2 2 1 1 27 + + 22 + 125 8 3 3 5 5 3 5 8 4 1 44 1 1 = 27 + + + 125 8 27 9 5 3 25 125 4 44 4 44 =8+ + + 1 8= + +1 45 75 45 75 20 + 132 + 225 377 = = = 1 152 225 225 225 a 1 + y b 2 + mx 4 8. x 2y 2m+ 3y2xb2x 4b2 2m x 43x a 1 + 2 y b 2 3mx 4 2 3 52 1 + 2 4 3 2 3 3 9.nb 1 mx 3+8 5nb 1 3m x 3 + 10 4nb 1 + 5m x 3 18 mx 3 2 3534 2 = = 3 9 254= 15 + 8 2 = 212 2 b 3 1 1 ab + b 2 3 9 1 1 ab b 2 6 3 ++ 2ab 2b3 + 83a 13 2 a 412.9 2 m 17 7 2 m 34 7 2 m 34+ +25 2 n 34 5 2 n 17 + +1 4 1 2 1 4 3 15mn 30mn18 + 7 7 2 25 + 10 2 1 + 2 1 + 12 45mn m + n + 34 34 4 18 2 35 2 12 = 45mn m + n + 34 34 4 9 2 35 2 m 45mn + n +3 17 34 2 2 9 2 2 1 35 1 = 45 + +3 17 3 3 5 34 5 9 4 90 35 1 + +3 17 9 15 34 25 4 35 4 7 = 6+ + 3= + 3 17 170 17 850 40 + 7 510 463 = = = 2 123 170 170 170 = 18. 13.1 2 b m 2 3 2 b m 4 1 2 b m 4 1 b 2m 8 + +3 cn 5 1 cn 10 1 cn 25 2cn2+14.+ 0,6ab2 0,3a 2b 0,8b 36+ 0,4a 34+0,2a 3 + 0,4ab2 0,5a 2b3 50,2a 0,8a 2b+6+ 15a 2b + 0,9b3 ,3ab 2 0,1a 2b + 0,1b 3 + 6 0,1a b + ab + 0,1b3 + 6 24 + 6 2 1 2 30 5 + 2 + 100 10 + 30 + 20 + 3 b m+ cn + 8 50 5 7 2 67 43 7 2 2 67 1 43 b m+ cn + = 3 + 10 + 8 50 5 8 3 50 5 5 7 9 2 67 43 126 670 43 21 67 43 = + + = + + = + + 24 50 5 24 250 5 4 25 5 525 + 268 + 860 1. 653 53 = = = 16 100 100 1002= 0,1 22 3 + 2 32 + 0,1 33 + 6 = 0,3 4 + 2 9 + 0,1 27 + 6 = 12 + 18 + 2,7 + 6 , = 25,5EJERCICIO 20 21. 8xa + 2 11= 8xa + 2 11 1. 8 5 = 1322.2. 7 4 = 11 3. 8 11 = 3 4. 8 ( 11) = 8 + 11 = 3( )5. 1 9 = 1 + 9 = 8 6. 2a 3b = 2a 3b 7. 3b 2 = 3b 2( 60a ) = 45a + 60a = 15a ( 86b ) = 54b + 86b = 140b ( 60m ) = 35m + 60m = 25m23. 45a x 1 24.54bn 125. 35ma 18. 4x 6b = 4x 6b 9. 5a 6b = 5a 6b( )10. 8 x 3 = 8 x + 3 11. 9a 5b2 = 9a 2 5b 2)27. n 1n 1aax 1n 1aa1 10 + 1 11 1 = =52 2 22 3 89 17 = = 3 4 12 1229.4 3 5 3 24 + 25 3 49 3 x y x y = x y= x y 6 5 30 301 2 3 2 1+ 6 2 5 2 ab ab = ab = ab 4 8 8 815. 6 x 2 y ( x 2 y) = 6x 2 y + x 2 y = 5x 2 y30. 16.11a 3m2 ( 7a 3m2 ) = 11a 3m2 + 7a 3m2 = 18a 3m231. 2 3 = 517. 8ab 2 ( 8ab 2 ) = 8ab 2 + 8ab 2 = 018.31x 2 y ( 46 x 2 y) = 31x 2 y + 46 x 2 y = 77 x 2 y19. 84a 2b ( 84a 2b) = 84a 2b + 84a 2b = 0( ) 33. 8 ( 5) = 8 + 5 = 3 34. 5 ( 4 ) = 5 + 4 = 9 35. 7 ( 7) = 7 + 7 = 0 36. 2a ( 5) = 2a + 5 32. 7 1 = 7 + 1 = 8x 1n 11 2 2 2 1 2 2 2 3 2 2 x x = x + x = x = x 3 3 3 3 314. 11 2 25 m2 = 14m2 m20. 3a x + 1 5b x + 2 = 3a x + 1 5b x + 2x 128.12. 7 xy 5 yz = 7 xy + 5 yz 13. 3a 4a = ax 126. 5 = 5 + = 2 22(6a n ( 5a n ) = 6a n + 5a n = 11a n 19. EJERCICIO 2137. 3x b = 3x b 38. 2n 5m = 2n 5m(40.(11a + b 4)( ) 44. 4b ( 3b) = 4b + 3b = b2 4. x 3x 5x 6x 2 + 2x 643. 3a a = 3a + a = 4a54 x 3 ( 11x 3 ) = 54 x 3 + 11x 3 = 65x 33 2 5. a a b 7a 2b 9ab 2 54a 2 y ( 43a 2 y ) = 54a 2 y + 43a 2 y = 11a 2 y48. ab 9ab = 10ab 31x y ( 31x y) = 31x y + 31x y = 0 22250. 3a a = 4a x51.x53. 31ax 18a54. 236ma x 1x 2 + y 2 3xy8. 3x 2 + y 2 + 4xy2 x + 2 y 2z9. 2x 2 + 2y 2 + xyx3 x2 + 6 5x 6 + 4 xx= 49a2x 6x 2x 1( 19m ) = 236m aa2 + 6y 3 8 10. y 2 4 6y 3y 2y4 y 2 2y 4 + 6y 3 6 y 8+ 4x3 2y 4 + 6 y 3 + 4 y 2 6y 8+ 19ma = 217ma 11. a 3 6ab 2 + 9a+ 8a 15a 2b 51 1 56. ( 6a ) = + 6a 4 4 2 + 15 13 2 = 57. ( 5) = 3 3 3a 3 6ab 2 + 17a 15a 2b 5 a 3 15a 2b 6ab 2 + 17a 5 12. x 4 + 9xy 3 11y 47 3 3 3 58. m m 10 8 56 30 3 86 3 43 3 m = m = m = 80 80 405 2 2 11 2 2 a b ab 12 6 10 + 11 2 2 21 2 2 7 2 2 = ab = ab = ab 12 12 4 1 3 2 a b 45a 3b 2 9 1 405 3 2 406 3 2 1 ab = a b = 45 9 a 3b2 = 9 960. 0 2x 2 + xy + 2y 255. 85a x + 2 54a x + 2 = 139a x + 259.x+yzx52. 105m 9m = 96m x 17. x + y z x+ y z2311a x +1 ( 7a x +1 ) = 311a x +1 + 7a x +1 = 318a x +1 xxy+z6.a 3 8a 2b 9ab 246. 78a 2b 14a 2b = 64a 2b49.3x 5y3. 8a + b 3a 442. 25ab 25 = 25 ab + 2547.x 2y341. 7a ( 9 ) = 7a + 945.2. 2x 3y2b8b ( 5a ) = 5a + 8b 3a +b a +b1.)39. 3b 6a = 6a + 3b 20y 4 + 8x 3y + 6x 2 y 2 x 4 + 9xy 3 31y 4 + 8x 3 y + 6x 2 y 2 x 4 + 8x 3y + 6x 2 y 2 + 9xy 3 31y 4 13. a + b + c d a + bc+d14.2a + 2b 15.x 3 + 6x 2 9x 19 6x 3 + 11x 2 21x + 43 5 x 3 + 17 x 2 30x + 24ab + 2ac 3cd 5de 8ab + 4ac + 5cd 5de 7 ab + 6ac + 2cd 10de 20. 5 3 2 16. y 9y + 6y 31 31y 3 + 8y 2 + 11y 4 + 19y y 5 40y 3 + 14 y 2 + 11y 4 + 19y 31 y 5 + 11y 4 40y 3 + 14 y 2 + 19y 3117.5m3 9n3 + 6m2n 8mn2 5m3+ 21m2n 14mn2 + 18 9n3 + 27m2n 22mn2 + 18 27m2n 22mn2 9n3 + 1818.4 x 3 y 19xy 3 + y 4 6x 2 y 2 25x 3 y + 51xy 3 32 x 2 y 2 + x 429x 3 y + 32 xy 3 + y 4 38x 2 y 2 + x 4 x 4 + 29 x 3 y 38x 2 y 2 + 32 xy 3 + y 4 6 4 2 2 4 19. m + m n 9m n + 19 + 30m2n4 + 61+ 13m3n3 16mn5m6 + m4n2 + 21 2n4 + 80 + 13m3n3 16mn5 m6 + m4n2 + 13m3n3 + 21m2n4 16mn5 + 80 m a 5b20.+ 11a b68a+ 6 a 3b 3 4 18 ab 5 + 11a b22+ 42 9b468a 6 a 5 b + 11a 4 b 2 + 6a 3b 3 + 11a 2b 4 18 ab 5 9b 6 + 42 2 4 3 5 21. 1 x + x x + 3x 6x24 + 30x 2 8x 4 15x+ x625 + 29 x 2 7 x 4 x 3 12x 6x 5 + x 6 x 6 6 x 5 7 x 4 x 3 + 29 x 2 12 x + 2522. 6x 2 y 3 + 8 x 5 23x 4 y + 80x 3 y 2 18 + 51x 4 y + 21x 3 y 2 80 + y 5 9xy 4 6x 2 y 3 + 8 x 5 + 28x 4 y + 101x 3 y 2 98 + y 5 9xy 4 8x 5 + 28x 4 y + 101x 3 y 2 6 x 2 y 3 9xy 4 + y 5 98 8m4n26 23. m+ 21m2n4 6mn5 14m n523m n+ 8+ 24mn5 8n6 + 143 3m6 + 23m5n 8m4n2 14m3n3 + 21 2n4 + 18mn5 8n6 + 22 m 7 5 2 24. x 8x + 16x 23x 15 51x+ 18 + 8 x 6 25 x 4 + 30x 3x 7 59x + 16x 5 23x 2 + 3 + 8x 6 25x 4 + 30x 3 x 7 + 8x 6 + 16x 5 25 x 4 + 30x 3 23x 2 59 x + 325. 9a 6 15a 4b 2 + 31a 2b 4 b 6 + 15a b 49a665335+ 31a b 4b 6 25a 5b 53 a 3b 3 + 9 ab 5 + 14 9a 6 25a 5b 53 a 3b 3 + 31a 2b 4 + 9 ab 5 4b 6 + 14 2ax + ax +1 ax + 226.+ 14 3 b 25a b 53 a b + 9 ab2 5a x + 6a x + 1 + a x + 2 4a x + 7a x + 14 21. a a 1 + 3ma 2 27. m m 5ma 2 3ma + 1 8ma 34ma5ma ma 1 2ma 2 3ma + 1 8ma 3 3ma + 1 + 5ma ma 1 2ma 2 8ma 3 m+4 7am + 2 8a m + 6am 1 28. a + 14a m + 2 + 8a m 1 + 5am + 3 + 11a m + 1a m + 4 + 7a m + 2 8am + 14a m 1 + 5am + 3 + 11a m + 1 am + 4 + 5am + 3 + 7am + 2 + 11a m + 1 8a m + 14a m 1 a+2 7 x a + 9x a 1 + 25x a 2 29. x 19x a 45x a 1 xa+2 26x 36x aa 1+ 11x a + 1 60x a 3 + 25xa2+ 11x a + 1 60x a 3 x a + 2 + 11x a + 1 26x a 36x a 1 + 25x a 2 60x a 3n+1 n2 + 8mn 3 19mn 5 30. m 6m 5mn 2 6mn 3 9mn 5 8mn mn 4 mn + 1 11mn 2 + 2mn 3 28mn 5 8mn mn 4 mn + 1 8mn 11mn 2 + 2mn 3 mn 4 28mn 5EJERCICIO 22 b a 1.7. a + 2b 3c a + b 2cb a 2b 2a 2a + 2b 2a + 3b 5c2. 2x + 3y x+y8. 3n + 4m + 5p + n m px + 4y 2n + 3m + 4p 3m 2n + 4p3. 7a + 5 5a b9. x + 3y 6z xy +z 2a + 5 b 2a b + 52x + 2y 5z4. x 2 + 6 x10. 5b 2 + 8ab + a 2 + 5x26b 2 ab 3a 2 2x + 6 + 5 x 2 x + 5 x + 6 22b 2 + 7ab 2a 2 2a 2 + 7ab + b 25. x 2 y + 5xy 2 + xy2 5m2 n2 + 6mn x311.x 2 y + 6xy 2 x 3 x 3 + x 2 y + 6 xy 26.7a 2b + 5ab 2 6a 2 b 2 6m2+ 9mn12. 8x 2 + 5x 4 + 8a 3+ x 6 + x3a b + 5ab + 8a 8a + a b + 5ab 2 m2 + n2 + 3mn3322 8x 2 + 6 x 10 + x 3 x 3 8x 2 + 6x 10 22. 13.6 3 2 23. y + y + y + 914m2 8n + 16 14m 9 m2 23y 33 8n + 7 m3 m3 8n + 7y 6 22y 3 + y 2 + 14 8y 4 + 15y 5 + 8y y 6 + 15y 5 8y 4 22y 3 + y 2 + 8y + 1414. 8ab + 5bc + 6cd ab + bc 6cd8 6 4 2 24. x x + 3x 5x 9 36 7x 7 5x 5 + 23x 3 51x7 ab + 6bcx 8 x 6 + 3x 4 5x 2 45 7x 7 5x 5 + 23x 3 51x3 2 3 15. a 9a b b x 8 7x 7 x 6 5x 5 + 3x 4 + 23x 3 5x 2 51x 45 25a 2b + b 3 + 8ab 2 a 3 34a 2b+ 8ab 216. 6x 8x y 6xy 3225. x 7 3x 5 y 2 + 35x 4 y 3 8x 2 y 5 + 60 + 60x 4 y 3 + x 2 y 52 xy 2 + 6y 3 4+ 5a x + 1d27. 8an + 16an 4 + 15an 2 + an 3 5an 2 an 3 8an 1 7an 15an + 16an 4 + 10an 219. x 5 8x 4 + 25x 2 + 15 15a 8a n 6x 2 + 7 6x 3 + 9x x 5 8x 4 + 19x 2 + 22 6x 3 + 9x+ 10a+ 9x a + 220. 3xy 4 8x 3 y 2 19y 5 + 18 5n 1n2 8an 1 + 16an 43 2 1 28. 15x a + + 5x a + 6x a + 41x a x 5 8x 4 6x 3 + 19x 2 + 9x + 22 25y a x + 2 + 6a x a x + 3 a x + 2 3a x + 1 + 6a x 55a 4 + 2a 3b + 8a 2b 2 45ab 3 + 5b 4 6xy2 5a x + 3 3a x + 1 5 a x + 2 + 6a x 40ab 3 + 6b 4 7a 3b + 8a 2b 2 5ab 3 b 444 3x+3 8a x + 1 5 26. a 16n + 19c + 14 d 19c d 16n + 14 4 3 18. 5a + 9a b5 2 x 7 3x 5 y 2 + 95x 4 y 3 90x 3 y 4 7x 2 y 5 + 50xy 6 y 7 + 60m2 9n + 11c + 14 m2 7n + 8c y 7 90x 3 y 4 + 50xy 6x 3x y + 95x y 7 x y + 60 y 7 90x 3 y 4 + 50xy 6 76x 3 8x 2 y 7xy 2 + 6y 3 417.+ 5 8y 4 + 15y 5 + 8y+ 18x a 1 31x a + 1 + x a + 415x a + 3 + 14x a + 2 6x a + 59x a 1 31x a + 1 + x a + 4 x +x y 52 3 x a + 4 + 15x a + 3 + 14x a + 2 31x a + 1 6x a + 59x a 1 9xy 4 8x 3 y 2 44y 5 + 18 x 5 + x 2 y 3 x 5 8x 3 y 2 + x 2 y 3 9xy 4 44y 5 + 18m 1 21am 2 + 26m 3 + 14am 5 29. 9a5am 1 12am 2x 3 6x 4 + 8x 2 9 + 15x21. 25x 3+ am + 8am 414am 1 33am 2 + 26m 3 + 14am 5 + am + 8am 4+ 18x 2 + 46 25x + 11x 5 am + 14am 1 33am 2 + 26m 3 + 8am 4 + 14am 5 24x 3 6x 4 + 26x 2 + 37 10x + 11x 5 11x 5 6x 4 24x 3 + 26x 2 10x + 37 26a 3b 25 22. a+ 8ab 4 b 5 + 6 8a 4b a 3b 2 + 15a 2b 3 + 45ab 4+8a 8a b 27a b + 15a b + 53ab b + 14 543 22 345 15mx + 330. m mx+4x+4+ 50mx + 1 14mx 6m x 1 + 8mx 2 + 23m 15mx+3x+2+ 23mx+2+ 6mx + 1 + 56mx +1+ mx 1 14m 5mx 1 + 8m x 2 x 23. EJERCICIO 232 11. x 1 xy y 211.x 2 1 xy y 2 x 2 xy y 2 1 a +1 a+2+63 12. a2. 0 a+8 5a 2b + 8ab 2 b 3 a 3 5a 2b + 8ab 2 b 3 + 6 a+8+ y33 13. x+ 5 x y 17 xy 293.+52x + 5 x y 17 xy + y + 5 3 3a a + 5 2 3a a 4 a 3a 4 22232 9 x y + 8x y + 15 xy 3+ 164.2 23x 9x 3 y + 8 x 2 y 2 + 15 xy 3 1 5xy + x 1642 5xy + x 214 14. x x 2 5xy+ b55 15. a+ 11a b 2a b 8a b + 4ab 45. 13 24a + 11a 4b 2a 2b3 8a 3b 2 + 4ab 4 + b5 5 a 3 + a 2b ab 2 1 a + a b ab a + a b ab + 1 32232 a 5 + 11a 4b 8a 3b2 2a 2b3 + 4ab 4 + b52+ x24 16. x6. x 3 5xx 3 + 8 x 2 y + 6xy 2+ 50 + 25x3x 4 5x 3 + x 2 + 25x + 502x 3 + 8x 2 y + 6 xy 26 17. y + y 413 7. a 9y 5 17y 4 + y 3 18y 2+ 8a 2b 6ab 2 + b 3y 6 + y 41 9y 5 17y 4 + y 3 18y 2a 3 + 8a 2b 6ab 2 + b 3 y 6 9y 5 17y 4 + y 3 18y 2 + y 41 + y48.6 18. a5x 3 y 7x 2 y 2 + 8 xy 3 + y 4 m4+ 15a b+ a 2b 4 17a b30 b + a c d a b c d + 30+ 14ab 5 + b 64 3 19. x + x 11x 5 x + x 2 + 34 x 4 + x 3 16x + x 2 + 34 x 4 + x 3 + x 2 16 x + 34 13 20. m10. 16 14 b + a c d3 3a 6 + 15a 5b + 9a 4b 2 17a 3b 3 + a 2b 4 + 14ab 5 + b 6 5m4 + 18am3 7a 2m2 a 3m + a 4 4m4 + 18am3 7a 2m2 a 3m + a 4+ 9a 4b 2 55x 3 y 7x 2 y 2 + 8 xy 39.2 3 m2n 7mn2 + 3n3 m3 m2n 7mn2 + 3n3 1 24. EJERCICIO 24 1.1 2 a 2 1 2 1 2 a + ab b 2 4 3 55 3 6. 6 m + +2 +1 2 1 2 a + ab b 2 4 3 5 3 2 1 2 2 = a + ab b 4 3 54 2 5 xy yz + 5 3 97.4 2 135 + 5 xy yz + 5 3 9 4 2 140 = xy yz + 5 3 9 4 2 = xy yz + 15 5 9 5 3 3 bc 3. 5 1 bc 63 2 ab + cd 4 918 5 3 2 bc + ab + cd 30 4 9 3 13 2 ab + bc + cd 4 30 91 2 a b 4. 2 3 4 2 1 a b + 5 9 2 58 6+2 1 a b+ 10 9 2 3 8 1 = a b+ 10 9 25 2 5. 9 x 3 2 y 8 1 2 5 3 y xy + 10 7 115 2 30 + 8 2 5 3 x y xy + 9 80 7 11 5 38 2 5 3 = x2 y xy + 9 80 7 11 5 5 19 2 3 x 2 xy y + 9 7 40 111 2 3 m n mn2 2 83 2 a + 7 5 2 a 141 3 ab b2 3 5 1 1 + ab 2 865 2 23 a + ab 14 6 1 2 1 3 1 a ab b 2 + 14 6 5 88.++5 3 10 + 9 3 1 2 3 m + n + m n mn2 6 45 2 8 5 3 1 2 3 19 3 2 m + m n mn + n 6 2 8 45152.2 3 n 9 1 3 n 53 2 x 8 3 2 x 55 xy 6 3 + xy 10+1 2 y 102y 215 + 24 2 50 + 18 1+ 20 2 x + xy y 40 60 10 39 2 68 21 2 39 2 17 21 2 = x + xy y = x + xy y 40 60 10 40 15 103 2 9. a + a++a7 2 9 a a 8 105 6 7 88 + 7 2 10 + 9 20 21 a a+ 8 10 24 15 2 19 1 3 =a + a a 8 10 24a3 +7 3 2 10 . m + 12 mn 5 mn2 94 3 n 7 n3+5 2 1 m n+ 21 821 20 4+7 3 5 2 1 mn2 n + m n+ 36 7 21 8 5 1 11 1 m3 + m2n + mn2 n3 + 21 36 7 8m3 + 25. 3 4 3 x + x3y 5 411. x45 3 xy 7 1 + xy 3 3 5 2 2 x y 82 4 y 3 5 4 y 6 +12.35 4 3 3 5 15 7 3 4 5 4 x + x y x2y 2 xy + y 5 4 8 21 6 2 3 5 8 3 1 4 = x 4 + x3y x2y 2 xy y 5 4 8 21 61 3 7 8 a+ b c + d 2 5 8 9 7 1 1 7 + b c + d 20 8 9 8 1 12 + 7 7 +1 8 +1 7 a+ b c+ d 2 20 8 9 8 1 19 8 9 7 b c+ d = a+ 2 20 8 9 8 1 19 7 = a+ b c+ d 2 20 8EJERCICIO 25 1.3 2 a 8 5 a2 66. 5 2 a b+ 85 a 6+9 20 2 5 11 2 5 a a= a a 24 6 24 6 2.8a 1 a 2+ +5 3 a 65 2 2 + 7 2 1+ 18 5 3 a a b+ ab 8 8 3 6 5 2 9 2 19 5 3 = a b + ab a 8 8 3 6 5 5 9 19 a 3 + a 2b + ab2 6 8 8 356b 3 b 51 2 1 ab 4 3 7 2 ab 6 816 1 30 + 3 15 33 a+ b5 = a+ b5 2 5 2 5 7.2 2 3. x + x y 6 3 7 2 x y 9 3x3 +4.a+ b2 3 20 49 2 2 3 + 2 m n+ mn + mn3 6 11 56 9 2 29 2 2 5 = m4 + m3n m n + mn3 6 11 56 9 c 2 c 32 1 4+3 3+2 1 7 5 a+ b c= a+ b c 2 4 3 2 4 35.2 5 m + n 3 6 m n1 mn3 6 3 2 + mn3 9 +m4 +67 2 1 x y 6 = x3 x2y 6 9 91 3 a + b 2 42 3 5 2 2 mn + mn 11 14 7 2 2 m4 mn 81 p 2 + p +23 56 1+ 2 1 1 3 m+ n+ p= m n+ p 3 6 2 3 6 28. 1 5 x 27 4 1 3 2 2 2 3 1 4 x y + x y + x y + xy 8 14 3 3 3 3 2 1 x y + xy 4 7 8 7 2 91 5 7 4 1 6 3 2 2 2 3 8 + 3 4 63 + 2 x x y+ x y + x y + xy 2 8 14 3 24 9 1 5 7 4 5 3 2 2 2 3 11 4 65 x x y x y + x y + xy = 2 8 14 3 24 9 1 5 7 4 5 3 2 2 2 3 11 4 x x y x y + x y + xy 7 2 = 9 2 8 14 3 24 26. 7 5 x y+ 92 4 2 1 2 6 x y x 3y 3 x 2 y 4 + xy 5 + y 3 8 13 7 4 2 1 2 4 + x y x y xy 5 + y6 9 119. x6 x6 +10.7 5 6 + 7 4 2 1 3 3 11+ 1 2 4 xy+ x y x y x y 9 9 8 111 3 7 2 x x y 3 9 2 3 1 2 x + x y 3 65 2 xy 8 3 xy 2 4 +7 3 y 11+2 + 13 6 7 13 4 2 1 3 3 12 2 4 15 6 y = x6 + x5 y + x y x y x y + y 13 9 9 8 11 132 5 61+ 2 3 14 3 2 5 6 2 7 3 2 + 30 11 2 1 7 32 x x y+ xy y = x3 x y xy 2 y 3 3 18 8 11 5 18 8 11 511.3 4 2 3 2 4 5 6 mn mn + n 10 7 9 2 6 7 4 2 5 2 4 1 6 m + mn mn n 13 20 14 3+3 52 6 6 + 7 4 2 6 + 5 2 4 5 3 6 3 2 6 13 4 2 11 2 4 2 6 3 m + mn mn + n + = m + mn mn + n + 13 20 14 9 5 13 20 14 9 5 3 5 7 4 12. 8 c + 22 c d 5 4 + c d 117 3 2 1 2 3 1 5 c d + c d d 12 2 3 5 3 2 3 3 5 + cd 4 c d d 6 4 13 + 353 5 7 + 10 4 7 + 10 3 2 1 2 3 3 4 13 + 9 5 c + c d+ c d + c d cd d 35 8 22 12 2 4 39 3 17 4 17 3 2 1 2 3 3 4 22 5 = c5 + c d+ c d + c d cd d 35 8 22 12 2 4 39EJERCICIO 26 Para los problemas 1 al 14 las literales toman los siguientes valores: a = 1 b = 2 c = 3 x = 4 y = 5 m = 3/2 n = 2/5 2 1. a ab3 3 2. a + b+ 2b 3 + 5a 2b 6ab 2 3ab b 2 a 2 4ab b 2 = 12 4 1 2 2 2 = 1 8 4 = 11a + 3b3 + 5a 2b 6ab 2 a 3 + 5a 2b 6ab 2 + 3b 3 3= 13 + 5 12 2 6 1 22 + 3 23 = 1+ 5 2 6 4 + 3 8 = 1+ 10 24 + 24 = 11 27. 1 a 23.a22 2 4. 3m 5n m2 10n 2 8mn9. a2m2 15n 2 8mn 2m2 8mn 15n 2 2 a b + 5a b29 24 12 45 48 24 27 == = 2 5 5 10 10 18x 2 y 2+ 15y 4 316x ya a 4b2 + 5a 3b 3 3a 2b4 + b 5 5= 15 14 22 + 5 13 23 3 12 24 + 25 = 1 4 + 5 8 3 16 + 32 = 1 4 + 40 48 + 32 = 21 16ab + 10mn 8mx 3 3 2 = 16 1 2 + 10 8 4 2 2 5 = 32 + 6 48 = 74x 4 + 16x 3y 18x 2 y 2 + 6xy 3 + 6y 4 = 4 + 16 4 5 18 4 5 + 6 4 5 + 6 5 = 256 + 80 64 18 16 25 + 24 125 + 6 625 = 256 + 5.120 7.200 + 3.000 + 3.750 = 4.926 4322343 11. a3 2 3 6. a 7am + m 2 + 5am + 5m3 8a 2 m 11a 2b + 9ab 2 b 3 a 3 11a 2b + 9ab 2 b3a 3 2am2 + 6m3 8a 2 m a 3 8a 2 m 2am2 + 6m3 2= 13 1112 2 + 9 1 22 23 = 1 22 + 36 8 = 733 24 9 27 3 3 = 13 8 12 2 1 + 6 = 1 2 + 6 2 2 2 2 4 8 9 81 4 48 18 + 81 19 = 1 12 + = = 2 4 4 412. 7.3 310. ab + 10mn 8mx 15ab+ 6xy 9y 43 3a 2b 4 + b 55 4 23 2 9 48 4 3 2 = 2 8 15 = 2 3 2 5 2 5 4 10 54 5. x11 2 1 2 1 3 mn + mn + n + m3 6 4 2 4+1 2 3+ 1 2 5 m n+ mn + m3 = m3 + m2 n + mn2 6 4 6 3 2 2 3 5 3 2 3 2 27 5 18 3 4 = + + = + + 2 6 2 5 2 5 8 6 20 2 25 27 3 12 675 + 150 + 48 873 = + + = = 8 4 50 200 2001 2 1 5 1 1 5 a b+ c= a b + c 2 2 3 2 2 3 1 1 5 1 1 2 + 3 = 1 + 5 2 2 3 2 1 2 + 10 7 1 = = =3 2 2 2=32 2 3 8. 3 m n + 4 mn 2 n1 5 b+ c 2 32 2 7 1 a + ab b2 3 8 5 1 1 2 a 2 ab + b 6 10 4 1 2 7 8 2 1 2 1 2 1 1 a + ab b = a ab b 2 6 8 10 2 8 10 1 1 1 2 1 2 4 12 1 2 2 = 2 8 10 2 8 10 1 1 2 10 5 8 3 = = = 2 4 5 20 201 4 x 64 2 2 5 3 x x+ 3 6 81 4 2 2 5 3 x x x+ 64 3 6 8 1 4 2 2 5 3 = 4 4 4 + 64 3 6 8 32 20 3 96 256 80 + 9 = 4 + = 3 6 8 24 231 77 5 = = = 9 8 24 8 28. +x313.3 2 x y 163 3 x 42 2 xy 5 3 2 1 3 + xy + y 5 2514.43 3 3 2 23 2 1 3 x + x y xy + y 4 16 5 25 1 3 2 1 2 1 3 x3 + x y+ xy + y 4 16 5 25 1 3 2 1 1 3 = 43 + 4 5 + 4 52 + 5 4 16 5 25 1 15 4 125 = 64 + 16 + 25 + 4 16 5 25 = 16 + 15 + 20 + 5 = 562 x 1 x 5 a + a ax 3 + ax 2 5 6 ax 1 + 9a x 3 a x 2 2 5 x 1 x 5 54 x 3 a +a a 5 6 3 49 x 3 ax ax 1 + a 5 6 3 49 4 3 = 14 14 1 + 1 5 6 3 49 + 30 18 + 245 257 =1 + = = = 8 17 30 5 6 30 30EJERCICIO 27 + ab + b 21.a2 5b 2 a 2 ab + 4b 2a 2 + ab 4b 2 ab + 4b 2a2a+8 a+61 14 2a 3 5a 2 + 2a 22a 3 + 5a 2 2a + 2+ y3 a 3 5a 2 + 2a 35 x 3 9x 2 11x 7x 4 7x y 210.x 3 5x 2 y 4xy 2 y 3 x 3 12x 2 y 4xy 2 y 3+4 6x335a 2b 7ab 2 1111.12x 4 + 2x 3 + 9x 2 + 6x 57a ab 4n3 7a8a ab + b + 55m4 4n32+ ab 53212. 11n43+ 14n2 25n + 86a19n3 6n2 + 9n 4 7a 9b + 3ca 11n4 + 19n3 + 8n2 16n + 4a b +c5a a +bc 2a + b caa2a + b c+ b3a3 7a 35a b + 8ab + 65m428a + 9b 3c a1x411x 4 + 2x 3 + 9x 2 + 6x 4 11x 4 2x 3 9x 2 6x + 436.1a3 8a + 6 x 3 + 5x 2 y + 4xy 2 + y 35. 6a 2 24 x 2 5ax 3a 25a 2 + 6a 4 2a 3+ 4n3 25 x 26a 2 + 25x 2+ 4xy 23 2 3 4. 3m n + 4mn n 3 2 3m n 4mn + 5n3x 2 5ax + 3a 2 9ax + 7a 2 + 25 x 29.+ 5x 2 y09ax a 2 133 3. xmn+p m+npm n + p8.142.7. m + n p 2m 2n + 2p 7n3n5+ 4n11n 19n3 8n2 + 16n 4 4n5 + 11n4 26n3 8n2 + 20n 43+5223 29. 6a 3m13.+ 5am37a 5a m 11a m 43220. x 4 6x 2 y 2 + y 46 6m2+ 8x y + 31y 2 23223 8a 2m2a4 x 4 2x 2 y 2 32y 44x 4 + 2x 2 y 2 + 32y 47a 11a m 11a m + 5am 6m 6 4x 4 2x 2 y 2 + 32y 4 4045 4 21. 6n + n+ m4+ n2 + 7n3 n2 8n 6 7a 4 + 11a 3m + 11a 2m2 5am3 + 6m4 + 6 6n5 + n4 + 7n3 6a 4 + 11a 3m + 3a 2m2 5am3 + 7m4 + 6 8n 6 6n5 + n4 + 7n3 8n 6 4x 4 y14. 6xn+ 13x 2 y 3 9xy 4 + 8x y5+ xy3 24 2y 6x 4x y + 8x y + 13x y 8xy 2y 543 22 3 30x 3 y 2x54+ 40xy 4 + y 5+ 22a b + 10a 3b2 11ab 4 b5 a 5 + 22a 4b + 7a 3b 2 5ab 4 b 5 5a b 2a + b8x + 6y + 4 223.x 7 xy + 34y22x 2 7 xy + 43y 2 1618. 4a + 8ab 5b a 2 7ab + 6b 25a + ab + b 14x y + 5xy7a 2 11ab + b 2x 14x 2 y + 5xy 2 y 3+42b 2 82b 2 4+ 2b 225.3a 4b + 5c 7a + 8b 11 a + 2b 7c 5a + 6b 2c 11 a + b + 2c 6a + 7b5a 3 + 14a 2 19a + 826.+ 9a 1a5 x 14x y + 5xy y 3x 32222b 2 82333 7a 2 + 11ab + b 2 8 y3 24 5a + 6b 2c 11a219. x 3m 3m 5m + 6m 122 4a 2 ab + b 25a 2 + ab + b 224.5m4 3m 5m + 6m + 4x 2 7 xy + 34y 22 3m3 5m2 + 6m + 4+ 8m + 4329y 2 1624m3 5m2 2m 7m3 7 xy + 40y 22 b52 3a 5 + 17a 4b + 7a 3b 2 + 7a 2b3 5ab 4 2b 58 x + 6y + 6 6y 5ab 4 b 5 + 7a b48x + 6y + 4 17. x 3a 3b2 + 6ab 4a 5 + 22a 4b + 7a 3b26y 52342a+945 22. ab16. 8x557x 5 + 4x 4 y 38x 3 y 2 13x 2 y 3 + 48xy 4 + 3y 52a 19n 6n + 4n + 15n 8n 25 656x 5 + 4x 4 y 8x 3 y 2 13x 2 y 3 + 8xy 4 + 2y 515. a + b ab+ 3n4 + 8n36 a43 19y 3a a 54x 3 14x 2 y + 5xy 2 20y 3+ 3a 21+ 5a + 17a 10a + 6432 a 4 + 3a 35a 2a + 8a + 17a 10a + 1 5432 11 30. + 10m2n24 27. m+ 15n42 23m 11m n 4m n 433x y + 21x y + 18x y y 5 4 3mn + 16n2 2343 x 5 3x 4 y 21x 3 y 2 18x 2 y 3m 5m + 11m n + 11 n + 11mn 17n 432 23+ 4a 3b 25 28. a 7a b429x 4 y 47x 3 y 2+ 8ab 4 b5+ 15a b 25ab + 3b42 32 3x 5 + 32x 4 y 26x 3 y 2 + 18x 2 y 3 2xy 4 + y 5 m + 11m n + 4m2n2 + 3mn3 16n4 43 2x 5 + 3x 4 y + 21x 3 y 2 + 18x 2 y 3+ 7m2n2 + 8mn3 n46m4+ y55 29. x 11m n 14m n 3mn + n4 34 2xy 4 + y 5x x 1 30. 3a + 6a a x 7a x 1 + a x 25 a 3b2 + 3a 2b3 5ab 44a x a x 1 + a x 2a 5 7a 4b + 3a 3b2 + 18a 2b 3 22ab 4 + 2b 5 8a x + 2 7a x + 1 a x 3a 6a b 21ab52 364 a + 7a b 3a b 18a b + 22ab 2b 543 22 342a + 7a b 3a b 24a b + ab 543 22 34 4a5x+ 12 a x 1 + ax 1 ax 28a x + 2 7a x + 1 5a x + 13 a x 1 a x 2 2b 6 5EJERCICIO 28 +5 2x 62 1. xx4 x+6x 2 + 2x 1 22x + 2x 3x + 2x 1 22. 3a 5b + c a b 3c7a + b+ 3ax3 3a + b + c 3x 2 x + 16x x + 8 3x + 8 x + 510a 2 x 14ax 23x 2 8x 5 6x + 2x 2 8x + 32a +1 24.2aa3 + a2a 2 a4 + a5. ab + bc + ac 7bc + 8ac 9ab 6bc + 9ac 9+83+241a36x 3 x 2a 3 9a 2 x + 17ax 2 3x 3 a 3 + 9a 2 x 17ax 2 + 3x 33 +1 3. x 5x 3 x 2 + 7 39x + 43a 3 + a 2 x + 3ax 2 3x 3 7a + 7b + 3c7a 7b 3c 5a 2 x + 11ax 2 11x 3 a 4a 2 x + 6ax 2 + 8x 32a 3 + a 2 x + 3ax 2 3x 34a 6b 2c 8b 3c4a 6b 2ca2 3x 3a2 x6.a +a 3 a4x2xa4ab+ 9ac 5x + 4 x 22x x 4 x + x + 2ab 6bc + 9ac 9 4ab 9ac 3ab 6bc3 3x + 534 4 a4 + a3 + a2 a5ab 3bc + 4ac ab + 3bc + 5ac+ x24 7. x932 7 x 3 + 8x 2 3x + 4 x34x 7 x + 8 x 3x + 1 432x 4 x 3 4 x 2 + x + 2 x 4 + 7 x 3 8x 2 + 3x 1 x 4 + 6x 3 12x 2 + 4x + 12 31. n44 8. m17m n 4m n 7mn3 3 m42 2 m4+ 6m2n2+6 80n4 m2n2 + mn3 417m3n + 2m2n2 7mn3 81n417m3n + 2m2n2 7mn3 81n4 m4 m2n2 + mn3 + 2+ a39.a a 5+ 8 a 4 4a 3 + a 2m4 + 17m3n + 3m2n2 8mn3 81n4 2a 5 a 4 + a 3 11a 2 10a + 27 5a 2 11a + 2616a 3 8a 2 7a 15a 5 a 4 + a 3 11a 2 10a + 27 a 4 + 12a 3 7a 2 7a 15 y22 10. 3x 11xy + 9y 1419x 8xy + 9y 23x 2 11xy + 8y 2 14a 120x 2 11xy + 8y 2 3x 2 + 11xy 8y 2 + 142a 2 ab + b 212.3a 2 8ab + 7b 2 3a + 8 5a + 11ab 17ba 2 2a + 1 a 2 + 2ab 9b22 a 2 + 9ab + 3b 2 8ab 7b a 2 + ab 4b22a 2 2ab + 9b 2 a 2 + ab 4b 2 ab + 5b 2214 13. m m3 + 8m2 6m + 5 16m5 m 7m + 1 m4 m3 + 7m2 13m + 5m5 16m4 16m + 7m 3 m5 16m4 + 7m2 19214. x+ 1417 x 2a 40 11a 3 4a 2 3a + 42a520x 2 11xy + 8y 23a2 a +1+ a 4 12a 3 + 7a 2 + 7a + 15x 2 3xy y 2211.2+ a 7 6a 24+ m2n2 mn3m44+ 7m2 19 m + m 7m2 + 13m 5243m5 17m4 + m3+ 13m 24 y55 2x y + 5x y 7 x y 43 2 3y 52 3 7x y+ 6 xy3 27x 4 y x 3 y 2 + 11xy 4 84 xyx 2x y 2 x y 7 x y + 6xy 4y 8 515. a43 22 34+ 7a 46 3a4 5a3 2a + 8 3a 4+ 5a 4a + 1 25a a 6 3a 5 + a 4 5a 3 + 4a 2 14a + 13 20a x57x 4 y x 3 y 2+ 10xy 41 x + 9x y + x y + 7 x y + 4xy 4 + 4y 5 + 7 54+ 21a x 19ax3 22 3 7a + 41a 50a + 8a + 3a 5 a 4 + 5a 3 4a 2 + 14a 13 6a 6 + 8a 5 4a 4 2a 3 + 44a 2 44a+ 18a 3 x 2 4x 5 8 9a x 17a x + 11a x 4 7ax 4 + x 5 80 2 35a 5 3a 4 7a 3 + 48a 2 58a + 1324a 20a x + 2a x + 21a x 26ax + x 71 3 2+ 7a 2 8a + 5 3+ 9 2 3+ 9a 3 x 2 445a 5 3a 4 7a 3 + 48a 2 58a + 13 7a 3 x 2 453 2+4 6a 4 11a 35 16. a17 x y x y + 10xy 15 8a + 11a 3 a 25 x 5 + 2 x 4 y + 2 x 3 y 2 + 7 x 2 y 3 6 xy 4 + 4y 5 + 84453 22 3+ 36a5 a 9a x + a x 5a 5 9a 4 x + a 3 x 2 + 11a 2 x 343 2+ 11a x 4 x 4 x 5 + 28 a + 20a 4 x 2a 3 x 2 21a 2 x 3 + 26ax 4 x 5 + 71 511a 4 x a 3 x 2 10a 2 x 3 + 26ax 4 5x 5 + 992 35+ 28 32. EJERCICIO 29 71 b 1. a + 2 2 3 a + b 3 45. 12 a3 a 4 1 a 35 b 494 a 12 5 = a 1232 2+3 a+ b 3 4 1 5 = a+ b 3 45 b 4 5 b 43 a6 81 3 3 2 a + a 2 5 5 3 3 2 3 a a a+6 6 5 85 3 3 a3 + a2 + a 6 6 5 83+5 3 3 a+6 a 6 8 4 3 = a3 a + 6 3 85 3 a3 + a2 6 53.a +3b2 2 a b 5 3+652 9 2 a+ b+6 5 3 7 3 = a+ b+6 5 31 3 4. 3 x 3 a+ 5 1 a+ 51 3 6 1 2 2 1+ 30 x x x+ 3 14 9 5 1 3 5 2 2 31 = x x x+ 3 14 9 5 5 3 x 6 1 3 5 2 2 31 x + x + x 3 14 9 5 5+2 3 5 2 2 31 x + x + x 6 14 9 5 7 3 5 2 2 1 = x + x + x6 5 6 14 9 7 b +6 3 1 b 63 1 14 + 1 a+ b+6 5 6 2 5 = a+ b+6 5 23 2 1 + x 7 5 1 2 2 x x+6 14 93 3 2 2 a + a 6 7 57+9 4 3 3 2 2 a a + a 12 7 5 4 4 3 3 2 2 = a a + a 3 7 57 4 3 3 2 2 a a + a 6 12 7 56. 2.7 4 3 2 6 a a3 + a2 12 7 5 3 4 1 1 a+ a 4 5 341 x+ 2 2 1 y z 3 4 1 2 y z +3 9 55 2 y 9 5 1 5 13 x y+ z 3 2 9 201 6 1 5+8 x+ y z+3 2 9 20 1 5 13 z+3 = x+ y 2 9 201 13 2 + 15 x z + 2 20 5 1 13 17 z = x+ 2 20 57.3 2 3 a b + ab2 2 4 1 2 5 a b ab 2 8 61 18 1 a 5 3 1 17 a 5 3 b3 +2 3 b 312 1 2 9 10 2 3 2 3 a b+ ab b 8 12 3 11 1 1 ab 2 b 3 = a 2b 8 12 3 1 3 a 21 3 b 3 11 2 1 1 a b+ ab 2 + b 3 8 12 3 1 3 11 2 1 a + a b+ ab 2 2 8 121 2 8. a b 2 9 1 b 33 c 51 23 3 a b c 2 9 5 1 1 3 = a+ b c 2 9 52 1 b + c 3 5 5 1 b c 9 101 1 3 a+ b c 2 9 5 1 1 b c 9 1065 2 1 1 6 +1 b+ c a c 9 10 2 10 1 1 1 7 = b+ = a c c 9 10 2 10 33. 1 3 1 2 9. 3 a + 8 a 3 a2 51 5 3 1 a 4 101 2 a 4 1 3 29 2 a a 3 40+1 3 5 24 2 3 2 1 a + a a+ 3 40 4 10 1 3 19 2 3 1 = a a a+ 3 40 4 103 2 5 10. 5 x 6 xy 3 xy 22 2 y 9 1 y2 32 2 x 9 17 2 x 451 43 2 10 + 18 23 2 1 x xy + y + 5 12 9 4 3 2 28 1 2 1 xy y + = x 5 12 9 4 3 2 7 1 2 1 = x xy y + 5 3 9 4211. 7 a31 3 b 5 3 3 1 3 a 2b + ab 2 + b 4 8 10 2 3 3 2 3 2 1 3 a a b + ab 2 b 7 4 8 10 2 3 3 2 3 2 1 3 = a a b + ab b 7 4 8 1012.2 4 3 m + m3n 7 5 1 4 7 3 m mn 14 20 +2 1 a+ 3 4 1 81 3 a 3 1 a3 + 31 3 10 29 2 2 2 1 a + a a+ 3 40 3 8 1 3 19 2 2 1 = a a a+ 3 40 3 8++1 xy 9 22 xy 92 2 y 3 3 2 y 2+1 2 2 mn 3 2 2 2 mn 5 1 2 2 mn 41 mn3 4 + + 1 8 1 1089 10 8 a+ 12 80 1 2 1 1 = = a+ a+ 12 80 12 40 1 210 + 17 2 1 22 4+9 2 1 x + xy y 45 9 6 2 27 2 21 13 2 1 x xy y = 45 9 6 2 3 2 7 13 2 1 = x xy y 5 3 6 225 2 2 +1 2 3 3 2 + 5 a b+ ab b 4 8 2 10 3 2 3 2 3 3 7 = a b + ab b 4 8 2 102 a 3 3 a 41 2 1 2 1 a b + ab 2 4 5 5 1 2 3 1 a 2b + ab b 3 4 8 2 219 2 a 40 19 2 a + 403 2 7 13 2 x + xy + y 5 3 6 3 2 7 1 x xy y 2 5 3 91 2 1 + 4 +39 2 2 2 + 1 y + 18 4 =37 2 3 y + 4 183 2 3 3 a b + ab 2 b 3 4 8 2 2 3 3 1 3 a 3 + a 2b ab 2 + b 7 4 8 10 7 102 3 15 1 3 7 a b 7 10 10 2 3 14 3 7 2 3 7 3 7 = a = a b b 7 10 10 7 5 10 n4 5 4 n 3 2 4 n 34 + 1 4 12 7 3 20 24 + 15 2 2 1 35+2 4 m + m n+ m n mn3 n 14 20 60 4 3 5 4 5 3 11 2 2 1 0 = m + m n+ m n mn3 n4 14 20 60 4 3 5 4 1 3 11 2 2 1 3 = m + m n+ m n mn 14 4 60 45 4 2 n4 m 14 5 5 4 1 3 11 2 2 1 3 m m n m n + mn 14 4 60 4 1 3 11 2 2 1 2 m n m n + mn3 n4 4 60 4 5 34. 13.1 1 x+ y 2 3 3 y 41 z 6 2 z 51 m 4 1 m 2+5 +1 3 n+ 3 81 4+9 5 12 1 2 1 3 x+ y z+ m + n+ 2 12 30 4 3 8 1 13 7 1 1 3 = x+ y+ z m + n+ 2 12 30 4 3 85 4 a + 614.1 3 a 2 3 4 a 83 3 a 4 1 3 a 6++2 2 a 3 1 2 a 61 13 7 1 1 40 3 x y z+ m n + 2 12 30 4 3 8 1 13 7 1 1 37 = x y z+ m n + 2 12 30 4 3 8+39 a 405+3 83 a 5 3 a 81 13 7 1 1 x y z+ m n 2 12 30 4 32 3 3 11+11 4 a 24 a420 9 4 6 9 + 2 3 4 1 2 24 + 15 39 165 22 + 9 a + a a a+ 24 12 6 40 33 11 4 1 3 3 2 0 152 11 4 1 3 1 2 152 = = a a a a+ a a a + 24 12 6 40 33 24 12 2 331 3 1 2 a a + 12 2 1 3 + a 12152 33 3 8. 11 24 4 1 1216 99 a2 + a 24 2 264 . 13 4 1 2 1117 = a a + 24 2 264EJERCICIO 30 1.x3 x2+54. x + x + 3x 5 6 323x65.3 2 6. 4x + 5x 5x 2 5x 2 4 x + 8 4x 3Rta. a 8a b + 5ab 3b4Rta. m 4m2n2 3mn3 + 6n4 + 8 8a 2b + 5ab2 4b 3 283 9x + 6Rta.4x 3 + 5x 2 5x 221 1 x y4 2 3 1 1 x+ y 2 34 a 3 8a 2b + 5ab2 3b3 28. m4 + 4m2n2 + 3mn3 6n4 8 b333 3mn3 + 6n4 4m2n2 11Rta. 5a + 8ab b3 112m4+ b34a 3 + 8ab 2Rta. x 4 x x + 13 38x + 9 Rta. 5a 9b + 6c + 8x + 9 a3 a3x 522. 5a + 9b 6c 5a 9b + 6c + 8x + 93.3 2 3 7. 5a + 8ab b 11+8 x 3 + 4 x 2 + x 13Rta. x + x + 3x 11 3x 3 4x 2Rta.1 1 x y4 2 3 35. 9.5x 2 7 xy 8y 2n310. 9m + 8m n 5mn + n 5x + 7 xy + 8y + 1 232211.Rta. 5 x + 7xy + 8y + 122322 a + 5a 8 a 3 + 5a 8 Rta. 0310m 8m n + 5mn 2n20 33 9m + 8m n 5mn + 2n 31 223m3 Rta. 10m3 8m2n + 5mn2 2n3EJERCICIO 31 6. a + (a b) + ( a + b)1. x ( x y)= x x+ y =y11. x + y + ( x y + z ) ( x + y z)= a+ ab a+b = 2a a = a()2 2 2. x + 3x x + 5= x+ y+ x y+ z x y+ z = x y + 2z= a b + 2a a + b= 3x + 5= 3a a = 2a22223. a + b ( 2a + 3)= a + b + 2a 312. a (b + a ) + ( a + b) ( a + 2b)2 2 2 2 2 7. a + b + 2a a b= x 3x x + 5 2222= a b a a + b + a 2b = 2b22(8. 2a { x + a 1} {a + x 3}= 2a + x a + 1 a x + 3= 3a + b 3) [(= 4m + 2m + n= x + y x 2 xy y x + y= 6m + n= x 2 xy + y2225. 2 x + 3 y (4 x + 3 y)= 2 x + 3y 4 x 3 y = 2x22222][]{} (2 2 2 2 2 14. 8x + 2 xy + y x + xy 3 y x 3xy= 8x 2 xy + y + x xy + 3y x + 3xy 22222= 8x 2 + 4 y 2210. ( 5m + 6) + ( m + 5) 615. (a + b) + ( a b) ( b + a ) + (3a + b)= 5m + 6 m + 5 6 = 6m + 5= a b a b + b a + 3a + b =0EJERCICIO 32[]2. 3x x + y (2 x + y)= 2a + a a b= 3x x + y 2 x y1. 2a + a (a + b)[= 2a + a a b = 2a b]]2= 3x 2 + 3xy + 2 y 22 2 2 2 2 2 9. x + y x + 2 xy + y + x + y 2) [(= x + y + xy 2 x + 3xy + y xy 2= 1+ 3= 44. 4m ( 2m n))13. x 2 y 2 + xy + 2 x 2 + 3xy y 2 + xy[[= 3x x y + 2 x + y = 4x]][]3. 2m (m n) (m + n)[= 2m m n m n = 2m m + n + m + n = 2m + 2n]) 36. [() () (2 2 2 2 2 4. 4 x + x xy + 3 y + 2 xy 3x + y[= 4 x + x + xy 3 y + 2 xy + 3x y 22222][]11. ( a + b) + (a + b) ( 2a + 3b) + ( b + a b)[]= a b + a b + 2a 3b b + a b = a b a b + 2a 3b b + a b = 3a 7b= 4 x 2 x 2 + xy 3 y 2 + 2 xy + 3x 2 y 2 = 6 x 2 + 3xy 4 y 2 5. a +)]{[{( 2a + b) ( a + b c) + a}() (2 2 2 12. 7m m + 3n 5 n 3 + m{[)] } (2n + 3)]} 2n 3 3n + 5 n 3 + m } 2n 3= a + { 2a + b + a b + c + a}= 7m2 m2 + 3n 5 + n + 3 m2= a 2a + b + a b + c + a =a+c= 7m { m[2] [] []]= 7 m2 + 2 n 5= 4m 2m + n 3 + 4n 2m 1{[= 5n + 2{[= 2a + 4a b 4a + b a + b a(= 6a b { 4a b + a b + a})]7. 2 x + 5x 2 y + { x + y}[]= 2 x + 5x ( 2 y x + y )[= 2 x + 5x + 2 y + x y]=b( [[(8. x 7 xy + y + x + 3xy 2 y{2{])}14. 3x 5 y + 2 x + y (6 + x) ( x + y)= 2x + y{]}= 6a b 4a + b a + b a= 2 x 5x + 2 y + x y2]}13. 2a ( 4a + b) 4a + (b a ) ( b + a )= 4m 2m n + 3 4n 2m 1[2= 7m2 + m2 + 3n 5 + n + 3 m2 2n 36. 4m 2m + (n 3) + 4n (2m + 1)[22[2= x 2 7 xy + y 2 x 2 + 3xy 2 y 2 = x 2 { 7 xy y 2 x 2 + 3xy 2 y 2 }( []) = 3x (5 y + [ 2 x + y 6 x + x y] ) = 3x 5 y + 2 x + {y 6 x} + x y)] } ]}= 3x (5 y 2 x + y 6 x + x y) = 3x 5 y + 2 x y + 6 + x x + y = 5x 5y + 6= x 2 + 7 xy + y 2 + x 2 3xy + 2 y 2[= 2 x 2 + 4 xy + 3y 2[{}9. (a + b) + 3a + b 2a + b (a b) + 2a[]= a b + a + b { 2a + b a + b}[= a b + a + b + 2a b + a b]= a b a + b + 2a b + a b = a 2b}][= 6c 2a c + { a c 2a a c } + 2c[]]} = x + y {4 x + 2 y + [ x y x y ] } [= x + y {4 x + 2 y x y x y} = x + y 4x 2 y + x + y + x + y]= 6c + 2a + c + a + c + 2a + a + c 2c = 6a + 7c))} (n + 6)] = 3m n [ 2m + { m + (2m 2n + 5)} n 6] = 3m n [2m + { m + 2m 2n + 5 } n 6] = 3m n [2m m + 2m 2n + 5 n 6] ({]= 6c 2a c a c 2a a c + 2c)[ {((16. 3m + n 2m + m + 2m 2n 510. ( x + y ) 4 x + 2 y + x y ( x + y )= 3x + y{15. 6c (2a + c) + (a + c) 2a (a + c) + 2c= 3m n 2m + m 2m + 2n 5 + n + 6 = 6m + 2n + 1 37. {[)] = 2a + { [5b + 3a c + 2 ( a + b c 4) ] + a b} = 2a + { [5b + 3a c + 2 + a b + c + 4] + a b} ([ {() [ ( ) ( )] = [ x + { x y [ x + y z + x y ] y}] = [ x + { x y + x y + z x + y y}] = [x x 2 y + z ]23. x + x + y x + y z x + y y}17. 2a + 5b + (3a c) + 2 a + b (c + 4) ( a + b)= 2a + { 5b 3a + c 2 a + b c 4 + a b}= x + x + 2y z = 2y z= 2a 5b 3a + c 2 a + b c 4 + a b = a 5b 6[)] { = [ 3x + ( x 2 y + 3)] + { 2 x y x 3 + 2 x y} (}18. 3x + x (2 y 3) + (2 x + y ) + ( x 3) + 2 ( x + y)[]= 3x x 2 y + 3 2 x y x 3 + 2 x y = 3x + x + 2 y 3 4 x 2 y 1 =4[] {[][] []}19. ( a ) + ( a ) + b + c + ( c)[ ] { [ ]} = a + a + {b c + c} = [a] a + b c c) [ ( ) ] }] = [ a + { a + a b a + b c [a + b] }] = [ a + { a c a b}] = [ a a c a b ] = 3a + b + c () (EJERCICIO 33()2. x 3xy y + 6 2=b2= x 2 + ( 3xy y 2 + 6){[]} { [ ]} = { [ a b] } { + [ b + a] } a b20. (a + b) + ( b a) (a + b)= {a + b} {b + a} a b3. x 3 + 4 x 2 3x + 1= x 3 + ( 4 x 2 3x + 1)4. a 3 5a 2b + 3ab 2 b 3= a b b a a b= a 3 + ( 5a 2b + 3ab 2 b 3 )= 3a 3b}] ]} { [ ]} [ { = { [ a b + c] } {+ [ c + a b] } + [ { a b}]{[21. (a + b c) + (c a + b) + a + ( b)[]= a b+ c+ c a+ b+ a+ b = a + b + 2c[ {[ {24. a + a + a b a b + c a + b1. a b + c d = a + b+ c d= b c+ c= {a + b c} { c + a b} + a + b))}+ { (m + n) + ( 2n + 3)}] = [ 3m + { m (n m 4)} + { m n 2n + 3}] = [3m + { m n + m + 4} m n 2n + 3 ] = [3m m n + m + 4 m n 2n + 3]( (22. 3m + m n m + 4= 3m + m + n m 4 + m + n + 2n 3 = 2m + 4n 7}]4 3 2 5. x x + 2 x 2 x + 1= x 4 x 3 + ( 2 x 2 2 x + 1)6. 2 a + b c + d = 2a b + c d()3 2 7. x + x + 3x 4= x 3 ( x 2 3x + 4)8. x 3 5 x 2 y + 3 xy 2 y 3= x 3 (5x 2 y 3xy 2 + y 3 )2 2 2 9. a x 2 xy y= a 2 ( x 2 + 2 xy + y 2 )2 2 2 10. a + b 2bc c= a 2 ( b 2 + 2bc + c2 ) 38. EJERCICIO 34 1. x + 2 y + ( x y )[6. 2a + ( 3a + b)[2. 4m 2n + 3 ( m + n) + (2m n)[[(2= x2[22[2]([]22)]8. x 3 3x 2 + 4x 2)4. x 3 3x 2 + 4 x + 2 3x 2 x + 3[) (2= 2 x 2 3xy + ( y 2 + xy) ( x 2 + y 2 )) ] {3xy [( x xy) + y ] }3. x 3xy + x xy + y 2(7. 2 x + 3xy y + xy + x + y]= 4m 2n 3 + ( m + n) (2m n) 2]= 2a ( 3a + b)]= x 2 y (x y)= x 3 + 3x 2 + 4 x 2{]= x 3 3x 2 4 x + 2 + 3x + (2 x + 3){[ ( )] } = 2a [ 3b + { 2a + [a + (b a )] }]5. 2a + 3b 2a + a + b a9.}[m (3m + 2m + 3)]+ ( 2m + 3) = { [ m (3m + 2m + 3)] ( 2m + 3)} 4242EJERCICIO 35 17. ambn ab = am + 1bn + 1 1. 2 3 = 618. 5ambn 6a 2b3 x = 30am + 2bn + 3 x2. 4 8 = 3219. xmync xmync x = xm + myn + nc1+ x = x 2my 2nc1+ x3. 15 16 = 240 4. ab ab = a1+ 1b1+ 1 = a 2b2 5. 2x 2 3x = 6x 2 + 1 = 6x3 6. 4a 2b ab 2 = 4a 2 + 1b1+ 2 = 4a 3b320. mxna 6m2n = 6mx + 2na + 1EJERCICIO 367. 5x3 y xy 2 = 5x3 + 1y1+ 2 = 5x 4 y 31. am am + 1 = am + m + 1 = a 2m + 18. a 2b 3 3a 2 x = 3a 2 + 2b 3 x = 3a 4b3 x2. xa x a + 2 = x a + a + 2 = x 2a + 29. 4m2 5mn2p = 20m2 + 1n2p = 20m3n2p3. 4anb x ab x + 1 = 4an + 1b x + x + 1 = 4an + 1b 2 x + 110. 5a y 6x = 30a x y4. a n + 1b n + 2 a n + 2b n = a n + 1+ n + 2 bn + 2 + n = a 2n + 3b 2n + 211. x2 y3 4y3 z4 = 4x2 y 3 + 3 z4 = 4x2 y 6 z45. 3an + 4bn + 1 4an + 2bn + 3 = 12a 2n + 6b2n + 412. abc cd = abc1+ 1d = abc 2d6. 3x 2 y 3 4x m + 1y m + 2 = 12x 2 + m + 1y 3 + m + 2 = 12xm + 3 y m + 513. 15x 4 y3 16a 2 x 3 = 240a 2 x4 + 3 y 3 = 240a 2 x7 y 37. 4x a + 2b a + 4 5xa + 5ba + 1 = 20x 2a + 7b 2a + 514. 3a 2b3 4x 2 y = 12a 2b3 x 2 y8. ambnc amb 2n = am + mbn + 2nc = a 2mb3nc15. 3a 2bx 7b3 x 5 = 21a 2b1+ 3 x1+ 5 = 21a 2b 4 x 69. xm + 1y a + 2 4xm 3 y a 5 c 2 = 4x2m 2 y 2a 3c 216. 8m2n3 9a 2mx 4 = 72a 2m2 + 1n3 x 4 = 72a 2m3n3 x 410. 5manb 1c 7m2a 3nb 4 = 35m3a 3n2b 5c222 2 39. EJERCICIO 37 3 m 2 3 1 3 m+1 3 a ab3 = am + 1b 3 = a b 4 5 2 5 101 2 4 3 1 4 4 5 2 a a b = a 5b = a b = a 5b 2 5 2 5 10 58. 3 2 7 2 3 m n am 7 14 3 7 2 5 21 2 5 3 2 5 a m n= a m n= amn = 7 14 98 141.9.2 2 3 3 2 4 6 2 6 4 2 x y a x y = a x y = a2x 6y 4 3 5 15 510. 2. 3.4. 7.2 x m +1 3 x 1 m a b a b 9 5 2 1 2 x 1 2m + 1 2 2 x 1 2m + 1 b a b = a = 3 5 151 3 4 4 3 2 4 3 5 5 1 3 5 5 m n a m n = amn = amn 8 5 40 107 2 3 14 4 1 a bc = a 4bc 5. abc a = 8 7 56 4 6. 5 m n 3 a b ab 2c 6 10 1 1 1 = am + 1 bn + 2 c = am + 1bn + 2 c 2 2 43 3 4 5 3 1 x y a 2by 5 = a 2bx 3y 9 = a 2bx 3 y 9 5 6 6 211.2 x +1 x 3 2 44 x 3 2 a b c a b 11 7 8 4 = 2 a 2 x 2b x 1c 2 = a 2 x 2b x 1c 2 7 712. 1 3 3 m+1 1 m+1 a am = a = a 3 5 15 5EJERCICIO 383 m n 4 12 3m 2n 3 3m 2n a b a 2mbn = a b = a b 8 5 40 103 3 1 x a m 5a 2m am 5 10 15 2 + x 3 +1+ a 3 x+2 a+4 a m a m = = 50 108. 1. a 3a a 2 = 3a1+ 1+ 2 = 3a 4 2. 3x 2 x 3 y a 2 x = 3a 2 x 2 + 3 +1y = 3a 2 x 6 y 3. m2n 3m2 5mn3 = 15m2 + 2 + 1n1+ 3 = 15m5n49. 2a a 2 3a 3 4a = 24a1+ 2 + 3 + 1 = 24a74. 4a 2 5a 3 x 2 ay 2 = 20a 2 + 3 +1x 2 y 2 = 20a 6 x 2 y210. 3b 2 4a 3b ab 5a 2 x5. am 2ab 3a 2b x = 6am +1+ 2b1+ x = 6am + 3b x + 1 6.7.1 3 2 2 3 6 2 + 4 3 +1 1 x a x a 4m = a x m = a 6 x 4m 2 3 5 30 5 2 m 3 2 4 a a b 3a 4b x + 1 3 4 18 m + 2 + 4 4 + x + 1 3 a b = = a m + 6b x + 5 12 2= 60a 3 + 1+ 2 b 2 + 1+ 1x = 60a 6b 4 x11. a mb x a 2 2ab 3a 2 x = 6a m + 2 + 1+ 2b x + 1x = 6am + 5b x + 1x1 2 3 10 3 3 2 x y xy 2 x x y 2 5 3 4 6 2 + 1+ 3 + 2 1+ 2 + 1 3 8 4 y = x = x y 8 412. EJERCICIO 39 1.2. 8x 2 y 3y 23x 3 x 2 2x 6x + 2x 42ax 3316ax 5 y 6ax 3 y 22 3. x 4x + 3 2x 2x 3 + 8 x 2 6 x3 2 4. a 4a + 6a 3ab3a 4b 12a 3b + 18a 2b 40. 2 2 5. a 2ab + b ab13. 4x 2 a 3b + 2a 2b 2 ab 3 4 x 5 + 12 x 4 20x 3 + 24 x 25 3 6. x 6 x 8x 2 2 3a x4 3 2 2 14. a 6a x + 9a x 8 3 3bx3a 2 x 7 18a 2 x 5 24a 2 x 33a 4bx 3 18a 3bx 4 + 27a 2bx 5 24bx 3m4 3m2n2 + 7n47.n+ 3 3a n + 2 4a n + 1 an 15. a an x 2 4m3 x 4m7 x + 12m5n2 x 28m3n4 x a 2n + 3 x 2 + 3a 2n + 2 x 2 + 4a 2n + 1x 2 + a 2n x 23 2 2 8. x 4x y + 6 xy16.ax 3 y ax 6 y 4ax 5 y 2 + 6ax 4 y 3 a 5a b 8ab 39.2x 3 3x 2 + 5x 6x 4 6 x 3 + 8 x 2 7x + 5 3a 2 x 3 3a 2 x 7 + 18a 2 x 6 24a 2 x 5 + 21a 2 x 4 15a 2 x 323 2 2 3 17. 3x + 5x y 7 xy 4y 2 2 5a xy 4 a 4m 2 4a 7m2 + 20a 6bm2 + 32a 5b 2m2 15a 2 x 4 y 2 + 25a 2 x 3 y 3 35a 2 x 2 y 4 20a 2 xy 5m m 1 + am 2 10. a a 2a18.x a + 5 3x a + 4 + x a + 3 5x a + 1 2x 2 2a m + 1 + 2a m 1+ 1 2a m 2 +1 2x a + 7 + 6x a + 6 2 x a + 5 + 10x a + 3= 2a m + 1 + 2a m 2a m 119. a 8 3a 6b 2 + a 4b 4 3a 2b 6 + b 8m+1 + 3 xm x m 1 11. x 2m 3x 5a 3 y 2 5a11y 2 + 15a 9b 2 y 2 5a 7b 4 y 2 + 15a 5 b 6 y 2 5a 3b 8 y 23 x 3 m + 1 + 9 x 3m 3 x 3m 1m n m 1 n + 2 m2 n+4 + a m 3b n + 6 20. a b + 3a b a bm n m 1 n +1 m2 n+ 2 12. a b + a b a b4a mb 33a 2b4a 2 mb n + 3 + 12a 2 m 1b n + 5 4a 2 m 2b n + 7 + 4a 2 m 3b n + 93a m + 2bn + 1 + 3am + 1bn + 2 3a mbn + 3EJERCICIO 40 11. 2a2 b 32 2 a 5 2 1+ 2 4 2 a a b 10 15 1 3 4 2 = a a b 5 15232. 3 a 4 b 2 3 a b 34 6 3 1+ 1 a3 + 1b + a b 9 12 4 4 1 3 2 = a b+ a b 9 23.3 1 2 a b+ c 5 6 5 5 2 ac 3 15 1+ 1 2 5 10 2 + 1 a c + abc 2 ac 15 18 15 5 2 = a 2c 2 + abc 2 ac 3 18 3 24. 5a2 +1 2 ab b 2 3 93a 2 x 6 2+2 3 6 a x + a2 + 1bx a 2b 2 x 5 3 9 6 2 = a 4 x + a3bx a 2b 2 x 5 3 41. 5.1 2 1 2 1 2 1 2 a b + x y 2 3 4 5 5 a 2m 81 2 2 1 x xy y 2 3 5 4 3 3 y 25 2+ 2 5 2 2 5 2 2 5 2 2 a m+ a b m a mx + a my 16 24 32 40 5 4 5 2 2 5 2 2 1 2 2 = a m+ a b m a mx + a my 16 24 32 83 2 3 6 3 x y xy 3 + 1 y 2 + 3 6 10 8 1 3 3 = x 2 y 3 xy 4 y 5 2 5 86.8. 3a 5b + 6c 3 2 3 a x 109.9 2 + 1 3 15 2 3 18 2 3 a x + a bx a cx 10 10 10 9 3 3 3 2 3 9 2 3 = a x + a bx a cx 10 2 5 7.2 3 1 2 5 1 m + m n mn2 n3 3 2 6 9 3 2 3 mn 4 6 3 + 2 3 3 2 + 2 1+ 3 15 1+ 2 2 + 3 3 2 3 + 3 m n + m n m n mn 12 8 24 36 1 3 5 1 = m5n3 + m4 n4 m3n5 m2n6 2 8 8 122 6 1 4 2 3 2 4 1 6 x x y + x y y 5 3 5 10 5 a3x 4 y3 72 4 2 2 1 4 x x y + y 9 3 3 3 4 x y 7 6 4 +3 4 3 2 + 3 2 + 4 3 3 4 + 4 x y x y + x y 63 7 21 2 7 4 3 5 6 1 3 8 = x y x y + x y 21 7 710 3 6 + 4 3 5 3 4 + 4 2 + 3 15 3 2 + 4 4 + 3 5 3 4 6 + 3 a x y + a x y a x y + a x y 35 21 35 70 2 5 3 1 3 4 9 = a 3 x10 y 3 + a 3 x 8 y 5 a 3 x 6 y 7 + a x y 7 21 7 1410. EJERCICIO 41 1. a + 3 a 13. x + 5 x4a 2 + 3a a3x + 5xa 2 + 2a 3x + x 202. a 3 a +1a 2 3a a3 a 2a 3 22 4x 20 24. m 6 m55. x + 3 x+57. 3x 2 y2x + y6x 2 4 xyx 2 3x 5x + 15+ 3xy 2 y 2x 2 8x + 156x 2 xy 2 y 26. a 2 a 38.5x 4y 3x + 2ym2 6m 5m + 30a 2 + 2a + 3a + 6+ 10xy 8y 2m 11m + 30a 2 + 5a + 6 15x 2 + 22xy 8y 22 15x 2 + 12xy 42. 11. a + b 8a 4b9. 5a 7b a + 3b13. 9m + 8n 6m + 4n 8a 2 + 8ab5a 2 7ab + 15ab 21b 54m2 + 48mn+ 4ab 4b22 36mn + 32n2 8a 2 + 12ab 4b25a 2 + 8ab 21b2 54m2 + 12mn + 32n212. 6m 5n mn10. 7x 3 2x + 414. 7y 3 2y 116m2 5mn14x 2 6x 14y 2 6y+ 28x 12 6mn + 5n+ 77y + 3314x + 22x 12mn 6m 11 + 5n 14y + 71y + 3322222EJERCICIO 42 2 2 1. x + xy + y xy3 2 5. a + a a a 1a 4 + a3 a2x 3 + x 2 y + xy 2 x y xy y 22a a33 y3x3a 3 3a 2b + 3ab 2 b 3+ am3 am2 + am 2a2 2 10. 3a 5ab + 2b 4a 5b12a 3 20a 2b + 8ab 2 m4n2 m2n4 n6 15a 2b + 25ab 2 10b 3 n612a 3 35a 2b + 33ab 2 10b 3m6+ a b + 2ab + b 24 2 2 4 11. 5m 3m n + n 3m n2x 4 4 x 3 + 6x 2 2 xa 3 + 2a 2b + ab 2 2+ 7x 3y2 + 2 +x 2+3 9x 2 + x + 3+ 6y 3y 5+ 3mn4 + 3m n2 3 n515m5 5m4n 9m3n2 + 3m2n3 + 3mn4 n5a 4 + a3 + a23y 5 6y 3 + 5y 2 342 12. a + a + 1 a2 a 13 9. 3y 6y + 53 2 4. x 3x + 1 x+3 5m n22x 4 x 3 9m3n215m5+ 3x 6 x + 9x 3 33a 3 + 3a 2b + 3ab 2 + b 3 am 2aam43 2 7. x 2x + 3x 1 2x + 32 2 3. a + 2ab + b a +bx4+am6 + m4n2 + m2n4 a 2b + 2ab 2 b 3+ 3x 9xam4 am3 + am2 2am4 2 2 4 6. m + m n + n 2 2 m na 3 2a 2b + ab 232 2a 2 + aa42 2 2. a 2ab + b a bx 4 3x 33 2 9. m m + m 2 am + a a3 a2 a 12y + 10 a2 a 1+ 5y 2 12y + 10 a4 a 2 2a 1 43. 13. x 3 + 2x 2 x19. x 2 2xy + y 2x 2 2x + 5 x 2 + xy + 3y 2x + 2x x 54 x 4 + 2x 3 y x 2 y 23 2x 4 x + 2x 43+ x 3 y 2x 2 y 2 + xy 32+ 5x + 10x 5 x 3+ 3x 2 y 2 6xy 3 + 3y 42+ 12x 2 5xx5 x 4 + 3x 3 y2 20. n 2n + 13 2 2 14. m 3m n + 2mnm 2mn 8n 2n2 12n4 2n3 + n2m5 3m4n + 2m3n2 n2 + 2n 1 2m4n + 6m3n 2 4m2n 3 8m3n 2 + 24m2n 3 16mn 4 m 5m n 5 5xy 3 + 3y 4+ 20m n 16mn42 3n4 2n3+ 2n 143 2 2 21. a 3a b + 4ab15. x 2 + x + 1x2 x 1a 2b 2ab 2 10b 3x4 + x3 + x2a 5b 3a 4b 2 + 4a 3b 3 2a 4b 2 + 6a 3b 3 8a 2b 4 x3 x2 x 10a 3b 3 + 30a 2b 4 40ab 5 x2 x 1a 5b 5a 4b 2 x 2 2x 1x44 2 16. x 3x + 222. 8x 3 12x 2 y + 6xy 2 9y 3x 2 2x + 3 x2x + 3y 3x6 2x+ 2x4+ 6x5+ 3x x 6 2x 5 4x3 9x416x 4 24 x 3 y + 12x 2 y 2 18xy 322+64 y 4 6y 3 + 2y 2 8y+ m3 + m2 4m 1 m6 + m5 4m4+ m2 4m 1+ 10y 3 15y 2 + 5y 20 4y + 4y 3 13y 2 3y 20 43 2 3 24. a + 2ax + 3x3 18. a 5a + 22a 2 3ax x 2a a+5 2 2a 5 5a 3 + 2a 2+ 3a x23+ 4a 3 x 2 + 6a 2 x 3 +a x3 2 25a + 10 + 7a 2 27a + 10 6a 2 x 3 9ax 44+ 5a 2a4+ 5a 27y 42y + 5m6 + m5 4m4 m3a5 a4 24 x 2 y 216x 43 2 23. 2y 3y + y 4m3 + 1a+ 24x 3 y 36x 2 y 2 + 18xy 3 27y 4+ 6x 3 7 x 2 4 x + 617. m3 + m2 4m 1a5+ 22a 2b 4 40ab 5 2a + 3a x + 5a x 543 2 2ax 4 3x 5 11ax 4 3x 5 44. 2 30. y 2y + 1 y 4 2y 2 + 225. x 4 3x 3 y + 2x 2 y 2 + xy 3 x 2 xy y 2y 6 2y 5 + y 4 x 6 + 3x 5 y 2x 4 y 2 x 2 y 3 2y 4 + 4y 3 2y 2 x 5 y + 3x 4 y 2 2x 3 y 3 x 2 y 4+ 2y 2 4y + 2 x 4 y 2 + 3x 3 y 3 2x 2 y 4 xy 5 x 6 + 2x 5 yy 6 2y 5 y 4 + 4y 3 3x 2 y 4 xy 5 4y + 24 2 31. m 3m + 426. a 3 5a 2 + 2a 3a 2a 73m3 2m + 1a 6 5a 5 + 2a 4 3a 33m7 9m53 2m 2a 4 + 10a 3 4a 2 + 6a+ 6m3 +m 7a + 35a 14a + 21 3a 5a 622 3m2543+423 2 32. a + a a + 1 3 a + a 2 2a 14 3 2 27. m + m m + 3m 2m + 3 2m6 + m5 m4a6 + a 5 a 4+ 3m2 2m5 2m4 + 2m3+a + a 6m+ 3m4 + 3m3 3m2 m6 m5 8m43m 11m + m + 18m 3m 8m + 4 7+ 31a 8a + 215+ 12m35+ 5m354+ a3 a3 + a2 2a 4 2a 3 + 2a 2 2a+9 a 3 a2 + a 1 6m + 9a 6 + 2a 5 2a 4 3a 3 + 2a 2 a 13 2 2 3 33. 8x 12x y 6xy + y28. a 4 + a 3b 3a 2b 2 ab3 + b 43x 2 2xy + 4y 2a 2 2ab + b224x 5 36x 4 y 18x 3y 2 + 3x 2 y 3a 6 + a 5b 3a 4b 2 a 3b 3 + a 2b 4 2a b 2a b + 6a b + 2a b 2ab 54 23 32 4 16x 4 y + 24x 3 y 2 + 12x 2 y 3 2xy 45+ a b + a b 3a b ab + b 4 23 32 4a 6 a 5b 4a 4b 2 + 6a 3b 35+ 32x 3 y 2 48x 2 y 3 24xy 4 + 4y 5624x 5 52x 4 y + 38x 3 y 2 33x 2 y 3 26xy 4 + 4y 5 3ab5 + b64 3 2 34. 5a 4a + 2a 3a 129. x 4 x 3 y + x 2 y 2 xy 3 + y 4a 4 2a 2 + 2x 2 + xy 2y 2 x x y+ x y 654 2 x y3 35a 8 4a7 + 2a 6 3a 5 a 4+ x y2 4 10a 6 + 8a 5 4a 4 + 6a 3 + 2a 2+ x 5 y x 4 y 2 + x 3 y 3 x 2 y 4 + xy 5 2x y + 2x y 4 2x63 3 2x y + 2xy 2y 2 456 2x 4 y 2 + 2x 3 y 3 2x 2 y 4 + 3xy 5 2y 6+ 10a 4 8a 3 + 4a 2 6a 2 5a 8 4a7 8a 6 + 5a 5 + 5a 4 2a 3 + 6a 2 6a 2 45. 4 3 2 35. x x + x x + 140. 3a 5 6a 3 + 2a 2 3a + 2x 2x + 3x + 6a 4 3a 2 + 4a 5x7 x 6 + x 5 x 4 + x 33a 9 6a 7 + 2a 6 3a 5 + 2a 432 9a 7 2x 6 + 2x 5 2x 4 + 2x 3 2x 2+ 12a+ 3x 5 3x 4 + 3x 3 3x 2 + 3x3a 9 15a 7 + 14a 6+ 30a 3 10a 2 + 15a 105 28a 4 + 47a 3 28a 2 + 23a 10+ x 2 3x + 641. a + b c ab+c3 2 36. 3a + 2a 5a 4a + a 2a + 1 3 24a 4 + 8a 3 12a 2 + 8a 15a+ 6x 4 6x 3 + 6 x 2 6x + 6 x 7 3x 6 + 6x 5+ 18a 5 6a 4 + 9a 3 6a 2 62a 2 + ab ac3a 6 + 2a 5 5a 4 4a 3 ab+ 3a 5 + 2a 4 5a 3 4a 2 b 2 + bc + ac 6a 4 4a 3 + 10a 2 + 8a + 3a 3 + 2a 2 5a 4+ bc c 2 b 2 + 2bc c 2a23a 6 + 5a 5 9a 4 10a 3 + 8a 2 + 3a 442. x + 2y z xy+z37. 5y 4 3y 3 + 4y 2 + 2yy 4 3y 2 1x 2 + 2xy xz5y 8 3y 7 + 4 y 6 + 2y 5 xy 15y 6 + 9y 5 12y 4 6y 3+ xz 5y 4 + 3y 3 4y 2 2y x 2 + xy5y 8 3y 7 11y 6 + 11y 5 17y 4 3y 3 4y 2 2y4 3 2 2 4 38. m 2m n + 3m n 4n m7 + 2m6n 3m5n2+ 4m3n4+ 3m n 6m n + 9m n 5243+ m4n3 2m3n4+ 2xy+ 20mn6 + 3m2n539. x 6 3x 4 y 2 x 2 y 4 + y 6x 2x y + 3xy4x11 3x 9 y 2 x 7 y 4 + x 5 y 6 6yz + 10z 2 2x 2 + 5xy xz 3y 2 yz + 10z 244. x 2 xy xz + y 2 yz + z2 x +y+zx 3 x 2 y x 2z + xy 2 xyz + xz 2 + x2y 2x 9 y 2 + 6x 7 y 4 + 2x 5 y 6 2x 3 y 8 xy 2 xyz +x z 2+ 3x 7 y 4 9x 5 y 6 3x 3 y 8 + 3xy10 x11 5x 9 y 2 + 8x 7 y 4 6x 5 y 6 5x 3 y 8 + 3xy10 3y 2 + 5yz + 4 xz 4n7 m7 + 5m6n 14m5n2 + 20m4n3 13m3n4 9m2n5 + 20mn6 4n73 2 2y 2 + 3yz z2 2x 2 + 3xy 5xz 12m2n5 5m5n2 + 10m4n3 15m3n45+ 2yz z 243. 2x 3y + 5z x + y + 2z m3 + 3m2n 5mn2 + n3 6 2y 2 + yzx3 xyz xz 3xyz+ y 3 y 2 z + yz 2 + y 2z yz 2 + z 32+ y3+ z3 46. x x 1 x2 7. a + 3a 2aEJERCICIO 43ax ax 1 + a x+ 2 x +1 x 1. a a + a a +1ax+ 3ax +2+aa 2x + 3a 2 x 1 2a 2x 2 a 2 x 1 3a 2 x 2 + 2a 2 x 3 + a 2 x 2 + 3a 2 x 3 2a 2 x 4x +1+ a x + 2 a x +1 + a x ax2x+ 3+aa 2x + 2a 2 x 1 4a 2 x 2 + 5a 2 x 3 2a 2 x 4x a+4 ma + 3 2ma + 2 + ma + 1 8. m2. xn + 3 + 2xn + 2 + xn + 1 ma 1 + ma 2 + mx +x 2 m2a + 3 + m2a + 2 + 2m2a + 1 m2a x n + 5 + 2x n + 4 + x n + 3+ m2a + 2 m2a + 1 2m2a + m2a 1 x n + 4 + 2x n + 3 + x n + 2+ m2a + 1 m2a 2m2a 1 + m2a 2 x n + 5 + x n + 4 + 3 x n + 3 + xn + 2ma+ 2 a +1 a a 1 3. m + m m + m 2ma + 3 2ma + 2 + 2ma + 1 2ma + 3m+ 2m2a + 2xa 1 xa 2 xma + 4 + ma + 3 ma + 2 + ma + 1ma + 4 ma + 32a + 3+ 2m2a + 1 4m2a m2a 1 + m2a 29. x a 1 + 2x a 2 x a 3 + x a 4m2 2m + 3a+ 2a3+ 3ma +1 3m + 3m aa 1a3x 2a 2 + 2x 2a 3 x 2a 4 + x 2a 5 x 2a 3 2 x 2a 4 + x 2 a 5 x 2 a 6+ 6ma + 1 5ma + 3ma 1 x 2 a 4 2 x 2 a 5 + x 2 a 6 x 2a 7 x 2a 2 + x 2a 3 4 x 2a 4n+ 2 n +1 n 4. a + 3a 2a n +1 n a +aa2n + 3+ 3a2n + 2 2an n 1 2 n 2 3 n3 4 10. a b a b + 2a b a b anb2 an 2b42n + 1+ a 2n + 2 + 3a 2n +1 2a 2na 2nb3 a 2n 1b 4 + 2a 2n 2b 5 a 2n 3b6a 2n + 3 + 4a 2n + 2 + a 2n + 1 2a 2n a 2n 2b 5 + a 2n 3b 6 2a 2n 4b7 + a 2n 5b 8 a 2nb3 a 2n 1b 4 + a 2n 2b55. x a + 2 + 2x a +1 x ax a + 3 2x a + 111. a x + b xx 2 a + 5 + 2x 2 a + 4 x 2 a + 3 2x x2a + 52a + 3 4x2a + 2+ 2x2a + 1+ 2x 2 a + 4 3x 2a + 3 4 x 2 a + 2 + 2x 2 a + 1a m + bm am + x + amb x + a xbm + bm + x a6. a 2a xx 1+ 3aa+ 2a+ 3ax +1 4a x + 6a x 1x a + 2a xax+2x 1 2a + 8a x+ a b + a xbm + bm + x m xx 1 n1 12. a b abx +1 2am+ xx2a 2 + 2a 1 x+2a x abn 1 3ax 1 x 2a 7x2 3ax2 a x 1b + bn a ab xn 1 a x 1b + bn 2a 2n 4b7 + a 2n 5b8 47. 2 2 2 1 2 13. 5a m + + a m + + 3a m6a 3m 1 8a 3m 2 + a 3m 3 30a 5m + 1 + 6a 5m + 18a 5m 1 + 40a 5m 8a 5m 1 24a 5m 2 5a 5m 1 + a 5m 2+ 3a 5m 3 30a 5m + 1 + 46a 5m + 5a 5m 1 23a 5m 2 + 3a 5m 3a + 2 x 1 a +1 x a x +1 14. x y 4x y + 3x y 2x 2a 1y x 2 4x 2a 2 y x 1 10x 2a 3 y x 2x 3a + 1y 2 x 3 + 8x 3a y 2 x 2 6x 3a 1y 2x 1 4x 3a y 2 x 2 + 16x 3a 1y 2 x 1 12x 3a 2 y 2x 10x 3a 1y 2 x 1 + 40x 3a 2 y 2 x 30x 3a 3 y 2 x + 1 2x 3a + 1y 2 x 3 + 4x 3a y 2 x 2+ 28x 3a 2 y 2 x 30x 3a 3 y 2x + 1EJERCICIO 44 3. 1.1 1 a b 2 3 1 1 a+ b 3 22 3 x 62 2 x y 9 3 2 x y 41 2 1 a ab 6 9 1 1 + ab b2 4 62 y 5 1 5 x+ y 3 6 1 2 x 3 +4.2 xy 15 5 10 2 xy y 6 301 2 4 25 10 2 x xy y 3 30 30 1 21 1 = x2 + xy y 2 3 30 3 1 7 1 = x2 + xy y 2 3 10 32 xy 2 12 3 2 3 + xy y 3 6 8+2 3 8 + 27 2 2+6 2 3 3 x x y+ xy y 6 36 12 8 1 3 35 2 8 3 3 2 = x x y+ xy y 3 36 12 8 1 3 35 2 2 2 3 3 = x x y + xy y 3 36 3 81 2 49 1 a ab b 2 6 36 6 1 5 1 = a2 + ab b2 6 36 62. x 1 2 1 1 x xy + y 2 2 3 4 2 3 x y 3 21 2 2 a ab + b2 4 3 1 3 a b 4 2 1 3 a 161 2 2 ab + ab2 4 12 3 2 3 2 ab + ab 8 26 3 b 61 3 2+3 2 2 + 18 2 a a b+ ab b3 16 8 12 1 3 5 2 20 2 = a a b+ ab b3 16 8 12 1 3 5 2 5 = a a b + ab 2 b 16 8 3 48. 5.2 2 1 1 m + mn n2 5 3 2 3 2 m mn + 2n2 2 6 4 3 3 m + m n 10 6 2 m3n 56.3 2 2 m n 4 1 m2n2 3 4 + m2n2 56 5 2 4 4 3 x + x x 8 4 5 3 3 x 241 mn 3 2 2 + mn 3 3+132 2 7. 2 x + 3 ax + 2 a8.3 2 2 x ax + a 2 2 3 3 4 3 3 x + ax 4 6 1 + ax 3 2+ 9 2 2 a x 4 1 2 2 a x 3 2 2 2 a x 63 a3x 2 2 + a3x 96 4 a 63 4 3 + 3 3 27 4 4 2 2 27 4 3 6 x + ax + a x a x + a4 4 6 12 18 6 3 6 19 2 2 23 3 6 = x 4 + ax 3 + a x a x + a4 4 6 12 18 6 3 4 19 2 2 23 3 3 4 = x + ax + a x a x+a 4 12 189.2 x 15 2 + x 4+2 3 1 x x2y + 7 5 1 2 2 x xy + 4 3 + 3 3 x 8 1 3 x 30 1 3 x 20+ 3 2 x 4 1 2 x 40 1 2 x 154 51 2 xy 2 5 2 y 6 1 3 2 x y 8 2 3 2 + x y 15 10 3 2 + x y 42+2 2 3 x y 6 5 2 3 x y 30 +5 xy 4 122 5 21+ 80 4 105 + 112 + 200 3 2 10 + 5 2 3 5 x x y+ x y x y + xy 4 28 420 840 30 12 2 5 101 4 417 3 2 15 2 3 5 = x x y+ x y x y + xy 4 28 420 840 30 12 1 5 101 4 139 3 2 1 2 3 5 x x y+ x y x y + xy 4 = 12 14 420 280 21 3 1 2 1 1 x + x x+ 4 3 4 2 3 2 1 1 x + x 2 10 5 3 5 3 4 x + x 8 6 1 4 + x 406 5 2 4 96 + 15 3 2 18 2 8 + 30 4 x + x x x + x 8 4 120 24 60 5 3 5 1 4 111 3 16 2 38 4 = x + x x + x + x 4 2 120 24 60 5 3 5 1 4 37 3 2 2 19 4 = x + x x + x + x 4 2 40 3 30 52 5 1 4 x x y 28 20 4 4 x y 21+1 2 x 12 6 2 + x 82 4 n 26 4 15 12 3 45 + 20 48 2 2 3 + 4 2 m + m n m n + mn3 n4 10 30 60 6 2 6 4 3 3 17 2 2 7 = m + m n m n + mn 3 n 4 10 30 60 6 3 4 1 3 17 2 2 7 = m + m n m n + mn3 n4 5 10 60 613 2 1 2 x + x 8 4 5 1 3 2x x + 2 31 x 20 1 1 + x 20 10+3 5 60 + 3 4 45 4 + 6 3 90 3 8 2 1+ 1 1 x + x x + x + x 8 120 120 120 20 10 3 63 4 47 3 79 2 2 1 3 5 21 4 47 3 79 2 1 1 = x5 + = x + x x + x + x x x + x + x 8 120 120 120 20 10 8 40 120 120 10 10 49. 31213 2 2 3 10. 4 m 2 m n + 5 mn 4 n2 2 2 5 m mn + n2 3 3 2 6 5 2 4 m m n + 12 6 6 4 m n+ 12 +4 3 2 m n 15 2 3 2 m n 6 15 3 2 m n 8 2 2 3 m n 12 4 2 3 m n 15 5 2 3 m n 42 mn 4 12 10 + mn 4 10+5 5 n 86 5 4+6 4 32 + 40 + 225 3 2 10 + 16 + 75 2 3 20 + 120 5 m m n+ m n m n + mn4 n5 12 12 120 60 120 8 1 5 10 4 297 3 2 101 2 3 140 5 5 1 5 5 4 99 3 2 101 2 3 7 5 4 mn n = m m n + m n m n + mn 4 n5 = m m n+ m n m n + 2 12 120 60 120 8 2 6 40 60 6 8EJERCICIO 45 3 2 2 3 3 2 3 4. m 5m n + 6mn + n por m 4mn n 1 5 + 6 + 1 1 + 0 4 13 2 2 1. x x + x por x 1 1 1+ 1 1+ 0 11 5 + 6 + 1 4 + 20 24 4 1 + 5 6 11 1+ 1 1+ 1 1 1 1+ 0 + 1 11 5 + 2 + 20 19 10 1= x5 x 4 + x 2 x= m6 5m5n + 2m4n2 + 20m3n3 19m2n4 10mn5 n64 3 2 3 2 2. x + 3 x 5x + 8 por x 2x 7 1+ 3 5 + 0 + 8 1 2 + 0 74 2 4 2 5. x 8x + 3 por x + 6x 5 1+ 0 8 + 0 + 3 1+ 0 + 6 + 0 51+ 3 5 + 0 + 8 2 6 + 10 + 0 16 7 21+ 35 + 0 561+ 0 8 + 0 + 3 + 6 + 0 48 + 0 + 181+ 1 11+ 3 13 + 19 + 0 561+ 0 2 + 0 50 + 0 + 58 + 0 15= x 7 + x 6 11x 5 + 3x 4 13x 3 + 19x 2 56= x 8 2x 6 50x 4 + 58x 2 153. a + 3a b 2a b + 5ab b 1+ 3 2 + 5 1 1 2 + 1 432 23 5 + 0 + 40 + 0 154por a 2ab + b 221+ 0 7 + 0 + 9 + 0 + 23 + 0 52 + 0 + 42 + 0 201+ 1 7 + 12 13 + 7 1 = a + a b 7a b + 12a b 13a b + 7ab b 54 21+ 0 4 + 0 + 3 + 0 2 1+ 0 3 + 0 6 + 0 + 10 4 + 0 + 12 + 0 + 24 + 0 40 + 3 + 0 9 + 0 18 + 0 + 30 2 + 0 + 6 + 0 + 12 + 0 201+ 3 2 + 5 1 2 6 + 4 10 + 2 +1 + 3 2 + 5 1 66. a 6 3a 4 6a 2 + 10 por a 8 4a 6 + 3a 4 2a 2 1+ 0 3 + 0 6 + 0 + 103 32 456= a14 7a12 + 9a10 + 23a 8 52a 6 + 42a 4 20a 2 50. 7. x 9 4x 6 + 3x 3 2 por 3x 6 8x 3 + 1012 8 4 16 12 8 4 8. m 7m + 9m 15 por m 5m + 9m 4m + 31 7 + 9 151+ 0 + 0 4 + 0 + 0 + 3 + 0 + 0 2 3 + 0 + 0 8 + 0 + 0 + 101 5 + 9 4 + 3 1 7 + 93 + 0 + 0 12 + 0 + 0 + 9 + 0 + 0 6 8 + 0 + 0 + 32 + 0 + 0 24 + 0 + 0 + 16 + 10 + 0 + 0 40 + 0 + 0 + 30 + 0 + 0 20 4 + 28 36 + 60 + 3 21 + 27 453 + 0 + 0 20 + 0 + 0 + 51+ 0 + 0 70 + 0 + 0 + 46 + 0 + 0 20 = 3x15 20x12 + 51x 9 70x 6 + 46x 3 20 15 5 + 35 45 + 75 + 9 63 + 81 1351 12 + 53 127 + 187 192 + 87 45 = m28 12m24 + 53m20 127m16 + 187m12 192m8 + 87m4 459. x 5 3x 4 y 6x 3 y 2 4x 2 y 3 y 5 por 2x 2 + 4y 25 2 4 2 10. 6a 4a + 6a 2 por a 2a + a 76+0+04+62 1+ 0 2 + 1 71 3 6 4 + 0 1 2+0+46+0+ 0 4+ 6 2 12 + 0 + 0 + 8 12 + 42 6 12 8 + 0 2 + 4 12 24 16 + 0 4+ 6+ 0 +04 +6 2 42 + 0 + 0 + 28 42 + 142 6 8 20 24 18 + 0 46 + 0 12 + 2 36 + 6 16 + 38 44 + 14= 2x 7 6x 6 y 8x 5 y 2 20 x 4 y 3 24 x 3 y 4 18x 2 y 5 4y 7= 6a 9 12a 7 + 2a 6 36a 5 + 6a 4 16a 3 + 38a 2 44a + 1412. 3x 4 4x 3 y y 4 por x 3 5xy 2 + 3y 3 3 4 + 0 + 0 111. n 3n + 5n 8n + 4 por n 3n + 4 643421+ 0 3 + 5 + 0 8 + 41+ 0 5 + 31+ 0 3 + 0 + 43 4 + 0 + 0 1 15 + 20 + 0 + 0 + 51+ 0 3 + 5 + 0 8 + 4 3 + 0 + 9 15 + 0 + 24 12+ 9 12 + 0 + 0 3 3 4 15 + 29 13 + 0 + 5 3+ 4 + 0 12 + 20 + 0 32 + 16= 3x 7 4 x 6 y 15x 5 y 2 + 29x 4 y 3 13x 3 y 4 + 5 xy 6 3y 71+ 0 6 + 5 + 13 23 8 + 44 12 32 + 16= n10 6n8 + 5n7 + 13n6 23n5 8n 4 + 44n3 12n2 32n + 1613. x10 5x 6 y 4 + 3x 2 y 8 6y10 por x 6 4x 4 y 2 5x 2 y 4 + y 6 1+ 0 + 0 + 0 5 + 0 + 0 + 0 + 3 + 0 6 1+ 0 4 + 0 5 + 0 + 1 1+ 0 + 0 + 0 5 + 0 + 0 + 0 + 3 + 0 6 4 + 0 + 0 + 0 + 20 + 0 + 0 + 0 12 + 0 + 24 5 + 0 + 0 + 0 + 25 + 0 + 0 + 0 15 + 0 + 30 + 1 + 0+ 0 + 0 5 +0+ 0+0+ 3 +0 6 1+ 0 4 + 0 10 + 0 + 21+ 0 + 28 + 0 23 + 0 + 9 + 0 + 33 + 0 6 x16 4 x14 y 2 10x12 y 4 + 21x10 y 6 + 28x 8 y 8 23x 6 y10 + 9x 4 y12 + 33x 2 y14 6y16 m m1 m 3 por a 2 5 14. a 3a + 5a1 3 + 0 + 5 1+ 0 51 3 + 0 + 5 5 + 15 + 0 25 1 3 5 + 20 + 0 25 = a m + 2 3a m + 1 5a m + 20a m 1 25a m 3x+2 x +1 x 1 por 7a x + 3 + 6a x + 1 + a x 15. a 5a 7a1 5 + 0 7 7 + 0 + 6 +17 35 + 0 49 + 6 30 + 0 42 + 1 5+ 0 7 7 35 + 6 78 5 42 7 = 7a 2 x + 5 35a 2 x + 4 + 6a 2 x + 3 78a 2 x + 2 5a 2 x + 1 42a 2 x 7a 2 x 1 51. 5x a 6 x a 2 por 6x a + 1 4 x a + 2x a 1 + x a 2 1+ 0 5 + 0 6 6 4 + 2 +116. xa+22x + 2 3a 2 x + 1 a 2 x 5a 2 x 1 por 6a 3x + 1 5a 3x + 3a 3x 1 1 3 1 5 65+317. a6 + 0 30 + 0 36 4 + 0 + 20 + 0 + 24 + 2 + 0 10 + 0 126 18 6 30 5 + 15 + 5 + 25 + 3 9 3 15+ 1+ 0 5 + 0 66 23 + 12 34 + 22 156 4 28 + 21 46 + 19 12 6= 6a 5x + 3 23a 5 x + 2 + 12a 5 x +1 34a 5 x + 22a 5 x 1 15a 5 x 2= 6x 2a + 3 4 x 2a + 2 28x 2a + 1 + 21x 2a 46x 2a 1 + 19x 2a 2 12x 2a 3 6x 2a 4EJERCICIO 46()(()()(())()(2 2 6. a b a 2ab + b a + b2 2 2 4. x + 1 x 1 x + 1)1. 4 a + 5 a 3x2 + 1 x4 + x2a 3 2a 2b + ab2x +14a + 20 a3 4a 2 + 20a)a 2 2ab + b 2 a+b2 12a 60+ x2 + 1+ a 2b 2ab2 + b34a 2 + 8a 60x 4 + 2 x2 + 1a 3 a 2b ab 2 + b 3 abx2 1()(a 4 a 3b a 2b 2 + ab 3x 6 + 2 x 4 + x2)2. 3a x + 1 x 1 2 a 3b + a 2b 2 + ab3 b 4 x4 2 x 2 13a 2 x + 3a 2a 4 2a 3bx6 + x4 x2 1x1+ 2ab3 b 43a 2 x 2 + 3a 2 x 3a 2 x 3a 2()()(5. m m 4 m 6 3m + 2 3a 23a 2 x 2)m 4m 2()()(3. 2 a 3 a 1 a + 4)m 3 4 m2 6m2 + 24ma 1 2a 2 6a 2a + 6 2a 2 8a + 6 a+4 2a 3 8a 2 + 6a + 8a 32a + 24 22a3m 10m + 24m 3 26a + 2423m + 2 3m4 30m3 + 72m2 + 2m3 20m2 + 48m 3m4 28m3 + 52m2 + 48m)()()3x 6 x + 3x x1 3m 62a 6(7. 3x x 2 2 x + 1 x 1 x + 1 23x 4 6 x 3 + 3x 2 3 x 3 + 6 x 2 3x 3x 4 9 x 3 + 9 x 2 3x x+1 3x 5 9 x 4 + 9 x 3 3x 2 + 3x 4 9 x 3 + 9 x 2 3x 3x 5 6 x 4+ 6 x 2 3x 52. ()()(2 2 8. x x + 1 x + x 1 x 211. (x 3)(x + 4)( x 5)( x + 1))x2 x + 1x3 x+ 4x5 x+1x2 + x 1x 2 3xx 2 5x+ 4 x 12+ x3 x2 + xxx2 4 x 5x2 4x 5 x2 + x 1x 4 + x 3 12 x 2 x2 + 2x 1x4 x 2+ x 5x 2 + x 12x4 x3 + x2 4 x 3 4 x 2 + 48x 5x 2 5x + 60 x + 2x x53 2xx 4 3x 3 21x 2 + 43x + 602+ 2x 4x + 2422 2 2 12. (x 3)( x + 2 x + 1)( x 1)(x + 3) 2 x + 2x + 1 x2 3x 2 x x + 4 x 2 5x + 2 543x2 + 3x1x + 2x + x 4m m 1 m 1 9. (a 3)(a + 2)(a 1) m a 32x+ 3x 2 + 6 x + 3+a 3am 1x 7 + 2 x 6 + 4 x 5 + 6 x 4 + 3x 3 62a + a m2 m 1 3am 1 x 6 2 x 5 4 x 4 6 x 3 3x 26 3x 5 6 x 4 12 x 3 18 x 2 9 xa m 1 1 2a a a+ 3x 4 + 6 x 3 + 12 x 2 + 18 x + 9 +a3m 2 3a2m22 m 12 m 1 6am 1+ 3a +a3m 2 3a2m2 3ax +x x 7m 1m 1 2a + 66 x 4 9 x 3 9 x2 + 9 x + 95m 2a m + 6 a 3m 2 + a 2 m 1 3a 2 m 2 2a m 3a m 1 + 62 13. 9a (3a 2)(2a + 1)(a 1)(2a 1) 27a 3 18a 2 2a 1 2a + 1 a154a 4 36a 3()()()10. a a 1 a 2 a 3a2 a2a 2 a+ 27a 18a 354a 9a 18a 432 2a + 122a 3a + 1 22a 2 3a + 1a2108a 6 18a 5 36a 4a3 a2 162a 5 + 27a 4 + 54a 3 2a + 2a+ 54a 4 9a 3 18a 2a 3a + 2a108a 180a + 45a 4 45a 3 18a 223+3x 3 x 2 3x + 3x 3 x 2 3x + 3 2 m 1m2 m 1 3x3 x2x 4 + 2 x3 + 4 x2 + 6x + 3a m 1 + 2 2a32a3 a 4 3a 3 + 2a 2 3a 3 + 9a 2 6a a 4 6a 3 + 11a 2 6a65(x x +1 x +2 14. a a + ba a2 x +1+a b x)(ax +1 b x + 2 )b xx+2b b2 x + 2x +1 xa 3 x + 2b x + a 2 x + 1b 2 x + 2 a 2 x + 1b2 x + 2 a xb 3x + 4 a 3 x + 2b x a xb 3 x + 4 53. EJERCICIO 47(1. 4 ( x + 3) + 5( x + 2 )) ()2= 3( x 2 + xy + xy + y 2 ) 4( x 2 xy xy + y 2 ) + 3x 2 3 y 2) (= 3( x 2 + 2 xy + y 2 ) 4( x 2 2 xy + y 2 ) + 3x 2 3 y 2)2 2 2 2. 6 x + 4 3 x + 1 + 5 x + 2= 3x 2 + 6 xy + 3 y 2 4 x 2 + 8 xy 4 y 2 + 3x 2 3 y 2= 6x + 24 3x 3 + 5x + 10 2(2= 3( x + y )( x + y ) 4 ( x y )( x y ) + 3x 2 3 y 2= 4 x + 12 + 5x + 10 = 9 x + 22()11. 3 x + y 4 x y + 3x 2 3 y 222= 2 x 2 + 14 xy 4 y 2= 8x + 31 212. (m + n) (2m + n) + (m 4n) 23. a (a x ) + 3a( x + 2a ) a( x 3a )= m2 + 2mn + n 2 (4m2 + 4mn + n 2 ) + m2 8mn + 16n 2= 10a + ax 22 2 2 2 2 2 4. x ( y + 1) + y ( x + 1) 3x y= 2m2 6mn + 17n 2 4m2 4mn n 2 = 2m2 10mn + 16n 2= x 2 y 2 + x 2 + x 2 y 2 + y 2 3x 2 y 2 = x2 y2 + x2 + y2(22222= ax + x 2)()(= y 3 + 3y 2() ( )( ) = 5x + 10 ( x + 4 x + x + 4) 6 x22222a+b c a+ b c= x + 10 x 2 4 x x 4ab+ c ab+ ca 2 + ab ac2a 2 ab + ac+ ab= x 2 6x + 6+ b 2 bc ac8. (a + 5)(a 5) 3(a + 2)(a 2) + 5(a + 4) ab bc + c 2= a 2 + 5a 5 3 (a 2 4) = a 2 + 5a 5 3a 2 + 12+ b 2 bc + aca + 2ab 2ac + b 2bc + c 2= a 2 5a + 5a 25 3(a 2 2a + 2a 4) + 5a + 2022 bc + c 2a 2ab + 2ac + b 2 2bc + c 2 2a+b+ c a+b+ c a 2 + ab + ac= 2a 2 + 5a + 7+ ab9. (a + b)(4a 3b) (5a 2b)(3a + b) (a + b)(3a 6b)= 4a 2 + ab 3b 2 (15a 2 ab 2b 2 ) (3a 2 3ab 6b 2 ) = 4a 2 + ab 3b 2 15a 2 + ab + 2b 2 3a 2 + 3ab + 6b 2 = 14a 2 + 5ab + 5b 2) ( ) = (a + c)(a + c) (a c)(a c) = a + ac + ac + c (a ac ac + c ) 222214. (a + b c) + (a b + c) (a + b + c)7. 5 x + 2 x + 1 x + 4 6 x2) + a + 2 x ) (a x)(a x ) a 2 x (a ax ax + x )= 2 x 2 + x + 5ax a a 2 210. a + c a c) (= 3ax x 2 + x a a 2 + ax + ax x 2)= y2 + x2 y3 x2 y3 y3 + x2 y2 + y2 x2 y2 + y2() ( )( + 3ax + 3x (ax + 2 x= 4ax + x + 3x ax 2 x2 2 3 3 2 2 2 2 2 6. y + x y y x + 1 + y x + 1 y x 12(2= 3m4 + m3 + 3m2 n 2 2mn2()13. x a + x + 3x a + 1 x + 1 a + 2 x a x5. 4m 5mn + 3m (m + n ) 3m(m n ) = 4m3 5mn 2 + 3m4 + 3m2 n 2 3m3 + 3mn2 22= (m + n )(m + n ) (2m + n)(2m + n) + (m 4n)(m 4n)= a 2 ax + 3ax + 6a 2 ax + 3a 2322= a 2 + 2ac + c 2 a 2 + ac + ac c 2 = 4ac2+ b 2 + bc + ac+ bc + c 2a + 2ab + 2ac + b + 2bc + c 2 22a 2 + 2ab 2ac + b 2 2bc + c 2 a 2 2ab + 2ac + b 2 2bc + c 2 a 2 2ab 2ac b 2 2bc c 2 a 2 2ab 2ac + b 2 6bc + c 2 a 2 + b 2 + c 2 2ab 2ac 6bc 54. () () (2)22 2 2 15. x + x 3 x 2 + x + x x 3x + x 3x + x2x + x 32x2 + x 2222x 4 + x 3 3x 2x4 + x3 2x2+ x 3 + x 2 3x() (x 4 + 2 x 3 3x 2 4 x + 4+ xyx 4 + 2 x 3 5x 2 6 x + 9x 4 x 3 3x 2 x 3 + x 2 + 3x x 4 2 x 3 + 3x 2 + 4 x 4 3x 2 + 3x + 9x 4 2 x 3 5x 2 + 6 x + 9x 4 2 x 3 5x 2 + 6 x + 9x 4 2 x 3 7 x 2 + 4 x + 14[][]2 17. x + (2 x 3) 3x ( x + 1) + 4 x x+ xz[ ][ ] = [3x 3][2 x 1] + 4 x xx + 2 xy + 2 xz + y + 2 yz + z 2 y2x22+ y23x + 3xy+ 3y 223x 2 + 5xy + 2 xz + 5y 2 + 2 yz + z 2 3x 2 + 5y 2 + z 2 + 5xy + 2 xz + 2 yz[[ ][ = [ x + 2][x + 8]= 3x + 6 4 x 4 3x + 12 2 x 42= x 2 6 x + 16= 5x 5x + 3)][ ( ) ( )] = [m n (m + 2mn + n )][2m + 2n 3m + 3n] = [m n m 2mn n ][ m + 5n] = [ 2mn 2n ][ m + 5n] ) ()(19. m + n m n m + n m + n 2 m + n 3 m n 22220.[ []]][xy y ] 2= 2 x 3 y + 10x 2 y 2 10xy 3 + 2 y 4[ []= 3x 2 y + x 2 y 2 x 2 y 6 x 3]]= 4x 6y 3 = 4x + 6y + 3{)] } { 3x + 5 [ x + 2 x x ] } { 3x + 5 + x 2 x + x } { 4 x + 5 + x }[(4. 4 x 2 3x + 5 x + x 2 x]2. (a + b) 3 2a + b ( a + 2)[][2= 2 x 3 y + 2 x 2 y 2 + 8 x 2 y 2 8xy 3 2 xy 3 + 2 y 4[= x 3a + 2 x 2 = 3 x 3a 2[2]3. 3x 2 y + ( x 2 y) 2 ( x + y) 3(2 x + 1)1. x 3a + 2 ( x + 1)[23EJERCICIO 48= x 3a 2 x + 2][2= 2 x 2 + 8xy 2 y 2 xy y 2= 2m2 n 10mn2 + 2mn 2 10n 3 = 2m n 8mn 10n2= x + 2 xy + y 3x + 6xy 3 y222[22[(x + y) 3(x y) ][(x + y)(x y) + x (y x)]= x 2 + 2 xy + y 2 3 (x 2 2 xy + y 2 ) x 2 y 2 + xy x 222]]= x 2 8 x + 2 x + 162)(][18. 3 ( x + 2) 4 ( x + 1) 3( x + 4) 2 ( x + 2)= 6 x 2 3x 6 x + 3 + 4 x x 2[ xy y 2+ yz + z 22= x + 2 x 3 3x x 1 + 4 x x 22+ y 2 + yz x2x2 x 32x 2 + xyx 2 + 2 xy + 2 xz + y 2 + 2 yz + z 2x2 x 3[()x+ y x yx 2 + xy + xz 2x2 2x + 4x 4 + 2 x 3 5x 2 6 x + 9) (x+ y+ z x+ y+ z+ x3 + x2 2x 3 x 2 3x + 9)(22 2 16. x + y + z x + y x y + 3 x + xy + y]= a b 3 2a ab + 2b= a b 6a + 3ab 6b = 7a 7b + 3ab 7a + 3ab 7b= 4x2 = 4x2 = 4x2= 4x2 + 4x 5 x2 = 3x 2 + 4 x 5222] 55. {[{() }] = m 3m 3n + [ { ( 2m + n 2 3m + 3n 3) + m}] = 2m 3n + [ { ( 5m + 4n 5) + m}] = 2m 3n + [ {5m 4n + 5 + m}] = 2m 3n + [ {6m 4n + 5}]10. m 3(m + n) + 2m + n 2 3[m n + 1] + m)) ] } = 2a { 3x + 2 [ a + 3x 2 ( a + b 2 a )] } = 2a { 3x + 2 [ a + 3x + 2a 2b + 4 + 2a ] } = 2a { 3x + 2 [3a + 3x 2b + 4 ] } = 2a { 3x + 6a + 6 x 4b + 8}[((5. 2a 3x + 2 a + 3x 2 a + b 2 + a[[]= 2m 3n + 6m + 4n 5= 2a + 3x 6a 6 x + 4b 8 = 4a 3x + 4b 8 4a + 4b 3x 8= 2m 3n 6m + 4n 5 = 8m + n 5]6. a ( x + y ) 3( x y) + 2 ( x 2 y) 2 ( x y)[ ]= a x y 3x + 3 y + 2 x + 2 y + 2 x + 2 y[]{ [] } { [ (x + y)] } = 3x + 6 y + 2 { 4 [ 2 x 3x 3 y ] } { [ x y] } = 3x + 6 y + 2 { 4[ 5x 3 y] } {x + y}11. 3( x 2 y ) + 2 4 2 x 3( x + y)= a 4x + 2y + 2 x + 4y = a 4x + 2 y + 2x + 8y = a 2 x + 10 y{[= 3x + 6 y + 2 {20 x + 12 y} x y]}7. m (m + n) 3 2m + 2m + n + 2 ( 1 + n) (m + n 1){[]}= 4 x + 5 y + 40 x + 24 y= m m n 3 2m + 2m + n 2 + 2n m n + 1{[]}= 36 x + 29 y= n 3 2m + 3m + 2n 1 = n 3 { 2m 3m + 2n 1} = n 3 { 5m + 2n 1}{[] }12. 5 (a + b) 3 2a + 3b (a + b) + ( a b) + 2 ( a + b) a= n + 15m 6n + 3{{ {[ [] }= 5 a b 3 2a + 3b a b a b 2a + 2b a= 7n + 15m + 3 15m 7n + 3]} = 2a + 2b 3a 6b 4 {a 2b + 2 [ a + b 1 + 2a 2b] } = 5a 4b 4 {a 2b + 2 [a b 1] } [8. 2 (a b) 3(a + 2b) 4 a 2b + 2 a + b 1 + 2 (a b)] } = 5 { 2 a b + 18a 9b} = 5{16a 10b} = 5 a b 3 6a + 3b a= 80a 50b= 5a 4b 4 {a 2b + 2a 2b 2} = 5a 4b 4 {3a 4b 2}{[] } 4 { [ ( a b)] } = 3{ [ a + b] } 4 { [a + b] }13. 3 + ( a + b)= 5a 4b 12a + 16b + 8 = 17a + 12b + 8[}] = 5x 5y [2 x y + 2{ x + y 3 x + y + 1}] + 2 x = 3x 5y [2 x y + 2 { 2 x + 2 y 2}]= 3{a b} 4 { a b}{9. 5( x + y) 2 x y + 2 x + y 3 ( x y 1) + 2 x[ = 3x 5y [ 2 x + 3 y 4]]= 3x 5y 2 x y 4 x + 4 y 4 = 3x 5y + 2 x 3 y + 4 = x 8y + 4= 3a + 3b + 4a + 4b = a + 7b{]} [ ]} = {a + b 2 a + 2b + 3 { [2a + b 3a 3b + 3] } 3[ a 2 + 2a ] } = { a + 3b + 3 {a + 2b 3} 3a + 6}{[14. a + b 2 (a b ) + 3 2a + b 3 (a + b 1) 3 a + 2 ( 1 + a )= { 4a + 3b + 6 + 3a + 6b 9} = { a + 9b 3} = a 9b + 3 56. EJERCICIO 498 9 4 4 12. a b c 8c = 1 8 9 4 4 1 a b c = a 8b9 8 86 4 3 13. 16m n 5n = 16 6 4 3 16 mn = m6n 5 51. 24 8 = 3 2. 63 7 = 9 3. 5a 2 a = 5a 2 1 = 5a 4. 14a 3b 4 2ab 2 = 7a 3 1b4 2 = 7a 2b 2 5. a 3b 4 c a 3b 4 = a 3 3b 4 4 c = c 6. a b ab = a 22 1 1 1b=a7. 54x y z 6xy z = 9x2 1y 2 2z3 3 = 9x 2 2 32 38. 5m2n m2n = 5m2 2n1 1 = 5 9. 8a 2 x 3 8a 2 x 3 = a 2 2 x 3 3 = 1 2 10. xy 2y = xy 2 1 xy = 2 24 5 4 11. 5x y 6x y = 5 4 4 5 1 5 x y = y4 6 6EJERCICIO 50 1. am + 3 am + 2 = a 3. 3am 23 m 2 b 5 g 3 m 2 m + 5 3 3 m = a = a a 5 5 5b 6a2m 2bx4108 7 6 6 8 8 27 7 ab c a = 20 52 2 1 6 6 2 m n = m 3 316. a x a 2 = a x 2 17. 3a xbm ab 2 = 3a x 1bm 2 3 4 m n 18. 5a b c 6a b c = x m m n 19. a b 4a b = 5 m 3 n 4 1 1 5 a b c = a m 3bn 4 6 61 xm mn a b 4 3 53 5a x 3 x 2 3 ax x2 33 = m a xnx 2 20. 3m n x 5m n x = m n xa 2. 2x a + 4 x a + 2 = 2x a + 4 b+ 2 g= 2x a + 4 a 2 = 2x 2b g= am + 3 m 2 = a3 2 x2 n 5. 4a b 5a b =7. 5a2 6 6 15. 2m n 3mn =m+3 m+2 5am 5 =2m 1 x 37 6 8 6 8 14. 108a b c 20b c =4 x 2 3 n 2 4 x 5 n 2 a b = a b 5 54. x2n + 3 4xn + 4 = m + 3 m 1 8x 4 y 2 = 6. 7x y7 m + 3 4 m 1 2 7 m 1 m 3 x y = x y 8 8n 1 n +1 n 1 n +1 8. 4x y 5x y = 5 5 = a 2m 1 2m + 2 b x 3 x + 4 = ab 6 69. am + nb x + a amba = am + n mb x + a a = anb x1 2n + 3 b+ 4g 1 1 n = x 2n + 3 n 4 = xn 1 x 4 4 42 3 m n x 10. 5ab c 6a b c = 4 n 1 n + 1 n + 1 n 1 4 x y = 5 55 1 m 2 n 3 x a b c 6EJERCICIO 51 1 1 2 2 2 2 3 2 x = x = x 1. 2 3 2 4 3 3 3 3 4 2 15 3 a= a a b a b = 5 a 3 2b1 1 = 4 2. 5 5 20 4 5 2 2 5 3 1 3 12 5 xy z z = 3 xy 5 z 3 3 = xy = 4xy 5 1 3. 3 6 3 64.7 7 m n 3 28 m 1 n 2 7 m 1 n 2 a b ab 2 = 8 a m 1b n 2 = a b = a b 3 8 4 24 6 42 5. 2 x 4 y 5 2 = 9 x 4 y 5 = 2 x 4 y 5 = 1 x 4 y 5 2 18 9 9 136. 3m4n5p 6 m4np 5 = 1 m4 4n5 1p 6 5 = 9n4p 3 3 7 7 2 5 6 5 5 6 8 2 1 5 5 6 6 = 14 a = 7 a 7. 8 a b c 2 ab c = 5 a b c 40 20 22 2 x m 3 2 10 x 1 m 2 8. a b ab = 3 a x 1bm 2 = a b 3 3 5 9 5 57. 11. 2a x + 4bm 3 1 a 4b 3 = 2 a x + 4 4bm 3 3 = 4a xbm 6 3 1 2 3 3 5 3 x 8 c 3 d 5 x = 12 c 3 d 5 x = 1 c 3 d 5