Algebra de baldor aurelio baldor (solucionario) (madrid)

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  1. 1. EJERCICIO 1 1. Una deuda se expresa en sentido1. Una deuda se expresa en sentido1. Una deuda se expresa en sentido1. Una deuda se expresa en sentido1. Una deuda se expresa en sentido negativo. Luego inicalmente el estadonegativo. Luego inicalmente el estadonegativo. Luego inicalmente el estadonegativo. Luego inicalmente el estadonegativo. Luego inicalmente el estado econmico de Pedro es - 60 bs.econmico de Pedro es - 60 bs.econmico de Pedro es - 60 bs.econmico de Pedro es - 60 bs.econmico de Pedro es - 60 bs. Al recibir 320 bs. Aumenta su capitalAl recibir 320 bs. Aumenta su capitalAl recibir 320 bs. Aumenta su capitalAl recibir 320 bs. Aumenta su capitalAl recibir 320 bs. Aumenta su capital tantos bs. como ha recibido; entoncestantos bs. como ha recibido; entoncestantos bs. como ha recibido; entoncestantos bs. como ha recibido; entoncestantos bs. como ha recibido; entonces la operacin queda expresada comola operacin queda expresada comola operacin queda expresada comola operacin queda expresada comola operacin queda expresada como se indicase indicase indicase indicase indica + = +60 320 260 bs. 2.2.2.2.2. 1 170 1 515 345. . = sucres 3.3.3.3.3. 200 56 189 67+ = + $ 4.4.4.4.4. + = +665 1 178 2 280 437. . soles 5.5.5.5.5. 20 15 40 75 30 + = $ 6.6.6.6.6. + + = 67 72 16 2 9$ 7.7.7.7.7. 200 78 81 93 41 59 70 + = colones 8.8.8.8.8. + =45 66 79 200 10 0 EJERCICIO 2 1.1.1.1.1. 12 15 3 = ! 2.2.2.2.2. + = 3 8 6 1! 3.3.3.3.3. ( )15 3 15 3 18 = + = ! 4.4.4.4.4. ( ) + = + =8 5 8 5 13! 5.5.5.5.5. + + = 4 7 2 11 6! 6.6.6.6.6. + = + = + = + = + = + = + 8 4 1 8 4 4 7 8 4 2 8 8 0 8 8 4 5 8 20 12 11 ! ! ! am am am 7.7.7.7.7. = = = = + = + = + = + = + 1 2 2 1 4 5 10 1 2 3 1 6 7 11 1 2 3 3 1 1 6 3 4 12 1 2 3 3 3 1 6 9 2 2 ! ! ! ! am am am pm 8.8.8.8.8. + = 56 7 49! 9.9.9.9.9. ( ) + = + = 71 5 66 15 5 20 ! ! long lat. 10.10.10.10.10. 18 3 21 65 4 61 + =+ - =+ ! ! long lat. 11.11.11.11.11. - + =+75 135 60 aos EJERCICIO 3 1.1.1.1.1. + 32 16m m; 2.2.2.2.2. + 10 4m m; 3.3.3.3.3. 50 85 35 = m 4.4.4.4.4. = 6 11 66m 5.5.5.5.5. - =- =+ 8 6 48 9 6 54 m m 6.6.6.6.6. 400 2 800 400 3 1 200 = + = m corredor m yo. 7. Los 40 pies de longitud del poste7. Los 40 pies de longitud del poste7. Los 40 pies de longitud del poste7. Los 40 pies de longitud del poste7. Los 40 pies de longitud del poste se reparten asi: 15 pies que sobresalense reparten asi: 15 pies que sobresalense reparten asi: 15 pies que sobresalense reparten asi: 15 pies que sobresalense reparten asi: 15 pies que sobresalen - se asumen en sentido positivo- .- se asumen en sentido positivo- .- se asumen en sentido positivo- .- se asumen en sentido positivo- .- se asumen en sentido positivo- . 25 pies que se encuentran enterrados.25 pies que se encuentran enterrados.25 pies que se encuentran enterrados.25 pies que se encuentran enterrados.25 pies que se encuentran enterrados. - se asumen en sentido negativo - .- se asumen en sentido negativo - .- se asumen en sentido negativo - .- se asumen en sentido negativo - .- se asumen en sentido negativo - . Al introducir 3 pies ms, se adicionanAl introducir 3 pies ms, se adicionanAl introducir 3 pies ms, se adicionanAl introducir 3 pies ms, se adicionanAl introducir 3 pies ms, se adicionan a los que estan bajo el suelo y se debena los que estan bajo el suelo y se debena los que estan bajo el suelo y se debena los que estan bajo el suelo y se debena los que estan bajo el suelo y se deben descontar de los que estan por encima.descontar de los que estan por encima.descontar de los que estan por encima.descontar de los que estan por encima.descontar de los que estan por encima. Aritmticamente significa:Aritmticamente significa:Aritmticamente significa:Aritmticamente significa:Aritmticamente significa: 40 15 25 25 3 28 15 3 12 = + = + = + pies pies 8.8.8.8.8. 55 52 3 = + m 9.9.9.9.9. + = 32 15 17m 10.10.10.10.10. 35 47 12 = m 11.11.11.11.11. + = +39 56 17m 12.12.12.12.12. 90 58 36 4 = m 13.13.13.13.13. 72 30 1 72 30 42 72 30 2 72 60 12 72 30 3 72 90 18 72 30 4 72 120 48 = = + = = + = = = = m m m m 14.14.14.14.14. ( ) ( ) ( ) ( ) = + = = + = = + = + = + = + 120 60 1 120 60 60 120 60 2 120 120 0 120 60 3 120 180 60 120 60 4 120 240 120 Km Km Km
  2. 2. EJERCICIO 7. 1.1.1.1.1. x x x+ =2 3 2.2.2.2.2. 8 9 17a a a+ = 3.3.3.3.3. 11 9 20b b b+ = 4.4.4.4.4. = b b b5 6 5.5.5.5.5. = 8 9m m m 6.6.6.6.6. = 9 7 16m m m 7.7.7.7.7. 4 5 9a a ax x x + = 8.8.8.8.8. 6 8 141 1 1 a a ax x x+ + + + = 9.9.9.9.9. = + + + m m mx x x1 1 1 5 6 10.10.10.10.10. = 3 42 2 2 a a ax x x 11.11.11.11.11. 1 2 1 2 1 1 2 2 2 a a a a a+ = + = = 12.12.12.12.12. 3 5 1 10 6 10 7 10 ab ab ab ab ab+ = + = 13.13.13.13.13. 1 3 1 6 2 1 6 3 6 1 2 xy xy xy xy xy+ = + = = 14.14.14.14.14. = = = 1 5 4 5 1 4 5 5 5 xy xy xy xy 15.15.15.15.15. = = 5 6 1 8 20 3 24 23 24 2 2 2 2 a b a b a b a b 16.16.16.16.16. = = a a a a 7 8 8 7 8 15 8 17.17.17.17.17. 8 9 6 23a a a a+ + = 18.18.18.18.18. 15 20 36x x x x+ + = 19.19.19.19.19. = 7 8 9 24m m m m 20.20.20.20.20. = a b a b a b a b2 2 2 2 3 5 21.21.21.21.21. a a a ax x x x + + =3 8 12 22.22.22.22.22. = + + + + 5 3 5 131 1 1 1 a a a ax x x x 23.23.23.23.23. a a a a a+ + = + + = 1 2 2 3 6 3 4 6 13 6 24.24.24.24.24. = = x x x x x 2 3 1 6 6 4 1 6 11 6 25.25.25.25.25. 1 5 3 10 2 3 10 10 15 10 3 2 ax ax ax ax ax ax+ + = + + = = 26.26.26.26.26. = = 3 4 5 6 9 10 12 12 31 12 2 2 2 2 2 a x a x a x a x a x 27.27.27.27.27. 11 8 9 11 39a a a a a+ + + = 28.28.28.28.28. m m m m mx x x x x+ + + + + + + + =1 1 1 1 1 3 4 6 14 29.29.29.29.29. =x y x y x y x y x y2 2 2 2 2 8 9 20 38 30.30.30.30.30. = 3 5 6 9 23a a a a am m m m m 31.31.31.31.31. 1 2 1 4 1 8 4 2 1 8 8 15 8 a a a a a a+ + + = + + + = 32.32.32.32.32. 2 5 1 2 1 10 1 20 8 10 2 1 20 21 20 ax ax ax ax ax ax+ + + = + + + = 33.33.33.33.33. 0 5 0 6 0 7 0 8 2 6, , , , ,m m m m m+ + + = 34.34.34.34.34. = = = 1 7 1 14 1 28 4 2 1 28 28 35 28 5 4 ab ab ab ab ab ab ab 35.35.35.35.35. = = 2 3 1 6 1 9 1 12 24 6 4 3 36 37 36 3 3 3 3 3 3 x y x y x y x y x y x y 36.36.36.36.36. ab ab ab ab ab ab2 2 2 2 2 2 7 9 21 39+ + + + = 37.37.37.37.37. = m m m m m m8 7 3 20 38.38.38.38.38. = + + + + + + x x x x x xa a a a a a1 1 1 1 1 1 8 4 5 19 39.39.39.39.39. 1 2 1 3 1 4 1 5 1 6 30 20 15 12 10 60 87 60 29 20 a a a a a a a a + + + + = + + + + = = 40.40.40.40.40. = = 1 3 1 6 1 2 1 12 1 9 12 6 18 3 4 36 43 36 ab ab ab ab ab ab ab EJERCICIO 8 1.1.1.1.1. 8 6 2a a a = 2.2.2.2.2. 6 8 2a a a = 3.3.3.3.3. 9 15 6ab ab ab = 4.4.4.4.4. 15 9 6ab ab ab = 5.5.5.5.5. 2 2 0a a = 6.6.6.6.6. + =7 7 0b b 7.7.7.7.7. + =14 32 18xy xy xy
  3. 3. 8.8.8.8.8. + =25 32 72 2 2 x y x y x y 9.9.9.9.9. 40 51 113 3 3 x y x y x y = 10.10.10.10.10. + =m n m n m n2 2 2 6 5 11.11.11.11.11. + =15 40 25xy xy xy 12.12.12.12.12. 55 81 263 2 3 2 3 2 a b a b a b = 13.13.13.13.13. + =x y x y2 2 0 14.14.14.14.14. + =9 9 02 2 ab ab 15.15.15.15.15. 7 7 02 2 x y x y = 16.16.16.16.16. + =101 118 17mn mn mn 17.17.17.17.17. 502 405 97ab ab ab = 18.18.18.18.18. + = 1024 1018 6x x x 19.19.19.19.19. + =15 15 0ab ab 20.20.20.20.20. 1 2 3 4 2 3 4 1 4 a a a a = = 21.21.21.21.21. 3 4 1 2 3 2 4 1 4 a a a a = = 22.22.22.22.22. 5 6 5 12 10 5 12 5 12 2 2 2 2 a b a b a b a b = = 23.23.23.23.23. + = + = 4 7 9 14 8 9 14 1 14 2 2 2 2 x y x y x y x y 24.24.24.24.24. 3 8 5 4 3 10 8 7 8 am am am am = = 25.25.25.25.25. + = + = am am am am 3 5 5 3 5 2 5 26.26.26.26.26. 5 6 7 8 20 21 24 1 24 mn mn mn mn = = 27.27.27.27.27. + = + = a b a b a b a b2 2 2 23 11 11 3 11 8 11 28.28.28.28.28. 3 4 5 6 2 24 3 4 3 4 3 , , ,a b a b a b = 29.29.29.29.29. + =12 3 4 2 2, , ,yz yz yz 30.30.30.30.30. 4 2 2a a ax x x = 31.31.31.31.31. + =+ + 8 8 01 1 a ax x 32.32.32.32.32. 25 32 71 1 1 m m ma a a = 33.33.33.33.33. + =+ + x xa a1 1 0 34.34.34.34.34. + = + = 1 4 1 2 1 2 4 1 4 2 2 2 2 a a a am m m m 35.35.35.35.35. 5 6 7 12 10 7 12 3 12 1 4 1 1 1 1 1 a a a a am m m m m+ + + + + = = = 36.36.36.36.36. 4 1 3 12 1 3 11 3 2 2 2 2 a a a a = = 37.37.37.37.37. + = + = 5 3 4 20 3 4 17 4 mn mn mn mn 38.38.38.38.38. 8 25 172 3 2 3 2 3 a b a b a bx x x x x x+ + + + + + = 39.39.39.39.39. + = + = 7 8 7 8 8 1 8 a b a b a b a bm n m n m n m n 40.40.40.40.40. 0 85 0 5 0 35, , ,mxy mxy mxy = EJERCICIO 9 1.1.1.1.1. 9 3 5 11a a a a + = 2.2.2.2.2. + =8 9 0x x x 3.3.3.3.3. 12 23 5 16mn mn mn mn = 4.4.4.4.4. + =x x x19 18 0 5.5.5.5.5. 19 10 6 15m m m m + = 6.6.6.6.6. + =11 15 26 0ab ab ab 7.7.7.7.7. + = 5 9 35 31a a a aX X X X 8.8.8.8.8. + =+ + + 24 15 39 02 2 2 a a aX X X 9.9.9.9.9. 2 3 1 3 2 1 3 3 0 3 0y y y y y+ = + = = 10.10.10.10.10. + = + = 3 5 1 4 1 2 12 5 10 20 17 20 m m m m m 11.11.11.11.11. 3 8 1 4 3 2 8 8 3 8 2 2 2 2 2 a b a b a b a b a b+ = + = 12.12.12.12.12. + + =a a a a a8 9 15 13.13.13.13.13. 7 11 20 31 15ab ab ab ab ab + = 14.14.14.14.14. 25 50 11 14 02 2 2 2 x x x x + + = 15.15.15.15.15. + =xy xy xy xy xy8 19 40 12 16.16.16.16.16. 7 21 80 53ab ab ab ab ab+ = 17.17.17.17.17. + + =25 11 60 82 362 2 2 2 2 xy xy xy xy xy 18.18.18.18.18. + + =72 87 101 243 157ax ax ax ax ax 19.19.19.19.19. + =82 71 53 206 0bx bx bx bx 20.20.20.20.20. 105 464 58 301 03 3 3 3 a a a a + + =
  4. 4. 21.21.21.21.21. 1 2 1 3 1 4 1 5 30 20 15 12 60 13 60 x x x x x x + = + = 22.22.22.22.22. 1 3 1 3 1 6 1 12 4 4 2 1 12 1 12 y y y y y y + = + = 23.23.23.23.23. 3 5 1 6 1 3 18 5 10 30 30 7 30 2 2 2 2 2 2 a b a b a b a b a b a b + = + = 24.24.24.24.24. + = + = = 5 6 1 6 3 8 20 4 24 9 24 9 24 3 8 2 2 2 2 2 2 2 ab ab ab ab ab ab ab 25.25.25.25.25. + + = a a a a a a8 11 15 75 64 26.26.26.26.26. + + + =7 21 14 30 82 80c c c c c c 27.27.27.27.27. + + =mn mn mn mn mn mn14 31 20 28.28.28.28.28. a y a y a y a y a y2 2 2 2 2 7 93 51 48 0 + + = 29.29.29.29.29. + + + =a a a a a a a3 6 3 30.30.30.30.30. 1 2 2 3 7 6 1 2 3 4 7 3 6 6 3 6 1 2 x x x x x x x x + + = + + = = 31.31.31.31.31. + + + = + + + = = 2 3 4 1 4 5 6 48 18 6 24 20 24 20 24 5 6 x x x x x x x x 32.32.32.32.32. 7 30 41 9 73 0a a a a ax x x x x + = 33.33.33.33.33. + + =+ + + + + + a a a a a ax x x x x x1 1 1 1 1 1 7 11 20 26 34.34.34.34.34. a a a a a a a+ + + =6 20 150 80 31 88 35.35.35.35.35. + = 9 11 17 81 110 9b b b b b b b 36.36.36.36.36. + + + = a b a b a b a b a b a b a b2 2 2 2 2 2 2 15 85 131 39 162 37.37.37.37.37. 84 501 604 715 231 165 1340 2 2 2 2 2 2 2 m x m x m x m x m x m x m x + + = 38.38.38.38.38. 5 6 2 3 1 4 5 8 4 20 16 6 15 96 24 111 24 37 8 4 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 5 8 3 2 a b a b a b a b a b a b a b a b a b + + = + + = = = 39.39.39.39.39. 40 81 130 41 83 91 16 28a a a a a a a a + + + = 40.40.40.40.40. + + =21 52 60 84 31 23 0ab ab ab ab ab ab ab EJERCICIO 10 1.1.1.1.1. 7 6 9 4 7 6 13 9 4 13 13 13 a a b b a a a b b b a b + + = = = 2.2.2.2.2. a b c b c c a a a b b c c c + + = = + = = 2 0 0 2 0 0 3.3.3.3.3. 5 11 9 20 1 5 20 25 11 12 9 1 10 25 12 10 x y x y x x x y y y x y + + = = = = 4.4.4.4.4. + + = = = = + 6 8 5 6 11 6 6 13 8 7 5 11 6 13 7 6 m n m n m m m m m n n n m n 5.5.5.5.5. + + + + + = + = + = = a b b c a c b a a a b b b c c a 2 2 3 2 3 3 2 2 3 0 2 2 0 2 6.6.6.6.6. + + + + + + = + = = 81 19 30 6 80 25 81 80 0 19 6 25 0 30 30 x y z y x x y x x x y y y z z 7.7.7.7.7. 15 6 8 20 5 31 15 8 8 6 5 12 20 31 11 8 12 11 2 2 2 2 2 2 2 2 a ab a ab a ab a a a a ab ab ab ab a ab + + + = = = = 8.8.8.8.8. + + + + = + = = 3 4 6 81 114 31 3 6 31 21 4 81 114 30 21 30 a b a b b a a b a a a a a b b b b b a b 9.9.9.9.9. + + + + + = + = = 71 84 50 84 45 18 71 50 45 18 48 84 84 0 48 3 4 2 3 4 2 3 3 3 3 3 3 3 4 2 4 2 3 a b a b a b a b a b a b a b a b a b a b a b a b a b a b 10.10.10.10.10. + + + + + + = + = + = = = a b c a b c a b c a a a a b b b c c c a 8 2 2 19 2 3 3 3 3 2 3 2 2 3 0 2 3 0 8 19 3 14 2 14
  5. 5. 11.11.11.11.11. m mn m mn m m m m m m m m m m m m mn mn mn m m mn 2 2 3 2 2 3 3 3 3 2 2 2 2 2 3 2 71 14 65 115 6 6 7 14 115 129 71 65 6 7 129 6 + + + + = = = = + 12.12.12.12.12. x y x y x y x y x y x y x y x y y x y x y x y x y x y x y x y x y x y x y y x y x y y 4 3 2 2 4 2 3 2 3 2 4 3 4 4 4 4 3 2 3 2 3 2 3 2 2 2 3 4 3 2 3 8 10 7 9 21 50 8 21 14 7 7 0 10 9 50 31 14 7 31 + + + + + = + = = + = = + 13.13.13.13.13. 5 3 8 5 50 4 65 90 7 8 7 0 3 4 0 5 5 0 50 65 90 25 25 1 2 3 1 2 2 3 3 3 3 3 2 2 2 1 1 a b c a b b c c c c c b b b a a x x x x x x x x x x x x x x x x + + + + + + + + + + + + + + + + + + + + + + = + = = + = = 14.14.14.14.14. a x a x a x x x x x a a a a x a m m m m m m m m m m m m m m m m + + + + + + + + + + + + + + + + + + + + = + = + = = 2 3 2 3 2 3 3 3 3 3 2 2 2 2 3 2 5 8 3 5 6 5 5 5 3 5 8 6 3 3 15.15.15.15.15. 0 3 0 4 0 5 0 6 0 7 0 9 3 3 3 0 3 0 6 3 2 7 0 4 0 7 3 3 3 0 5 0 9 3 3 4 2 7 3 3 3 4 , , , , , , , , , , , , , , , , , , a b c a b c a b c a a a a b b b b c c c c a b c + + + + = = = = 16.16.16.16.16. 1 2 1 3 2 3 3 4 1 6 3 4 1 2 1 2 2 3 4 2 8 3 4 7 4 1 3 3 1 6 2 18 1 6 17 6 3 4 1 2 3 2 4 1 4 7 4 17 6 1 4 a b a b a b a a a a a b b b b b a b + + + + = + = = = = = = + 17.17.17.17.17. 3 5 2 1 10 1 3 2 2 3 5 1 10 2 6 1 20 10 13 10 2 1 3 2 6 1 6 3 1 3 13 10 1 3 2 2 2 2 2 2 2 2 2 m mn m mn mn m m m m m m mn mn mn mn mn m mn + + + = + = + = + = = 18.18.18.18.18. + + + + = + = = = + = + = = = 3 4 1 2 5 6 2 3 4 1 6 1 3 2 3 4 7 3 9 28 12 19 12 1 2 3 4 2 2 3 8 4 9 4 5 6 1 6 1 3 5 1 2 6 6 6 19 12 9 4 2 2 1 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a ab b a ab b b ab a a a a ab ab ab ab ab b b b b b b a ab b
  6. 6. 19.19.19.19.19. 0 4 31 3 8 0 6 2 5 0 2 1 4 6 0 4 2 5 2 2 5 0 3 8 0 2 3 16 8 14 8 0175 0 6 1 4 2 4 1 4 14 4 0 35 31 6 25 0175 0 35 25 2 2 3 2 2 3 2 2 2 2 2 2 2 2 3 3 3 3 3 2 3 , , , , , , , , , , , , , , x y xy y x y xy y x y x y x y xy xy xy xy xy y y y y y xy y + + + = = = = = + = + = = = = + 20.20.20.20.20. 3 25 7 50 3 5 1 25 0 2 1 5 3 25 3 5 0 2 3 15 5 25 13 25 7 50 1 25 1 5 7 2 10 50 1 50 13 25 1 50 1 2 1 2 1 2 1 1 1 1 1 2 2 2 2 2 1 2 a b a b a b a a a a a b b b b b a b m m m m m m m m m m m m m m m m m m + + + = + = + = + = = + , , EJERCICIO 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 = 1/4a = 1 b = 2 c = 3 m = 1/2 n = 1/3 p = 1/4a = 1 b = 2 c = 3 m = 1/2 n = 1/3 p = 1/4a = 1 b = 2 c = 3 m = 1/2 n = 1/3 p = 1/4a = 1 b = 2 c = 3 m = 1/2 n = 1/3 p = 1/4 1.1.1.1.1. 3 3 1 2 6ab= = 2.2.2.2.2. 5 5 1 2 3 5 8 3 1202 3 2 3 a b c= = = 3.3.3.3.3. b mn2 2 2 1 2 1 3 4 6 2 3 = = = 4.4.4.4.4. 24 24 1 2 1 3 1 4 6 1 27 1 4 6 108 1 18 2 3 2 3 m n p= = = = 5.5.5.5.5. 2 3 2 3 1 2 1 2 2 3 4 1 8 8 24 1 3 4 2 3 4 2 3 a b m = = = = 6.6.6.6.6. 7 12 7 12 3 1 4 1 2 189 12 1 16 1 2 189 384 63 128 3 2 3 2 c p m= = = = 7.7.7.7.7. m n pb c a = = = 1 2 1 3 1 4 1 4 1 27 1 4 1 432 2 3 8.8.8.8.8. 5 6 5 6 1 1 2 5 6 1 2 5 12 1 2 2 1 3 2 a mb c = = = 9.9.9.9.9. 2 2 2 3 4 9 36 62 2 bc = = = = 10.10.10.10.10. 4 12 4 1 2 12 2 3 2 216 2 6 1223 23 3 m bc = = = = 11.11.11.11.11. mn a b = = = =8 1 2 1 3 8 1 2 1 6 64 8 6 4 3 4 3 4 3 12.12.12.12.12. 4 3 4 1 3 2 3 4 18 2 9 a bc = = = 13.13.13.13.13. 5 5 2 1 2 1 3 1 4 20 1 4 1 12 5 1 12 60 2 2 2 2 b m np = = = = 14.14.14.14.14. 3 4 2 3 3 4 2 2 3 3 3 8 4 2 9 3 24 4 18 3 6 6 1 3 2 3 2 b c = = = = = 15.15.15.15.15. 2 2 1 2 1 3 1 1 9 1 1 3 32 2 m n = = = = 16.16.16.16.16. 24 2 24 1 2 1 3 2 1 3 1 4 24 6 2 1 9 1 16 4 2 1 144 4 2 1 12 4 1 6 24 2 2 2 2 mn n p = = = = = =
  7. 7. 17.17.17.17.17. 3 64 2 3 64 2 3 2 1 2 3 64 8 729 3 373 248 3 72 216 3 63 3 63 3 3 = = = = = b c m . 18.18.18.18.18. 3 5 3 2 125 3 5 1 1 4 2 3 2 125 2 1 2 3 5 4 4 3 2 250 2 3 5 1 3 2 125 3 5 3 2 5 3 5 15 2 6 75 2 25 2 3 2 3 3 3 = = = = = = = apb bm EJERCICIO 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 1.1.1.1.1. a ab b2 2 2 2 2 3 2 3 4 4 9 24 16 1 + = + = + = 2.2.2.2.2. c cd d2 2 2 2 2 1 3 2 1 3 1 2 1 2 1 9 1 3 1 4 4 12 9 36 25 36 + + = + + = + + = + + = 3.3.3.3.3. a c b d + = + = + = 3 1 3 4 1 2 9 8 17 4.4.4.4.4. c d m n + = + = + = + = 2 1 3 1 2 6 1 4 2 2 3 24 2 2 72 6 3 64 3 5.5.5.5.5. a b m2 2 2 2 2 2 3 2 6 3 3 4 2 6 6 9 3 16 2 36 6 3 8 6 1 + = + = + = + = 6.6.6.6.6. 3 5 1 2 2 3 5 1 3 1 2 4 2 1 2 1 5 2 1 1 10 5 5 4 5 c b d + = + = + = + = 7.7.7.7.7. ab n ac d bd m + = + = + = + = + = = 3 4 1 4 3 1 3 1 2 4 1 2 6 12 1 4 1 1 2 2 6 48 2 1 3 144 6 1 3 149 3 49 2 3 8.8.8.8.8. b n m+ + = + + = + + = + + = =6 4 1 4 6 6 2 1 2 6 4 1 12 2 17 2 8 1 2 9.9.9.9.9. c a d b n d3 16 8 1 3 3 3 1 2 16 4 1 4 8 1 2 1 3 3 1 2 16 1 4 2 1 8 1 2 2 16 1 2 13 2 62 2 1 2 + = + = + = + = + = = 10.10.10.10.10. m d a b = = = = 6 1 2 216 1 16 216 16 3456 3 4 .
  8. 8. 111111.1.1.1.1. 3 4 4 3 1 3 4 4 1 4 6 1 3 4 1 4 6 1 12 1 24 2 1 24 3 24 1 8 2 2 2 2 c n m + = + = + = + = + = = 12.12.12.12.12. 4 2 16 2 1 2 1 2 8 1 4 1 2 4 8 16 1 1 2 1 2 1 1 1 0 2 2 2 2 d n + = + = + = + = = 13.13.13.13.13. a b c b m d + + = + + = = = 3 4 1 3 4 6 1 2 7 1 3 10 1 2 21 20 1 14.14.14.14.14. b a n m b d a + + = + + = + + = + + =5 4 3 1 4 6 4 1 2 5 3 1 1 4 2 1 2 15 4 4 15 23 15.15.15.15.15. 12 2 16 1 12 1 3 3 2 4 16 1 4 3 6 1 1 2 4 3 8 4 3 6 2 1 8 1 6 2 3 4 48 24 47 24 123 24 c a b n a m d + = + = + = + = + = = 16.16.16.16.16. 4 3 3 6 6 4 4 3 3 3 6 6 6 16 9 3 36 6 4 3 3 6 6 4 1 1 4b a m + = + = + = + = + = 17.17.17.17.17. b d c d+ + = + + = + + = = = 2 2 3 8 4 4 2 1 2 2 3 1 3 8 1 2 4 2 1 2 1 2 4 3 2 3 4 6 3 4 3 4 18.18.18.18.18. 2 3 3 2 4 2 3 4 3 3 2 1 2 4 3 1 4 2 144 3 3 9 4 4 3 1 2 2 12 3 3 3 2 4 3 2 24 3 9 2 4 3 2 8 9 8 3 2 64 9 12 8 61 8 7 2 2 2 2 2 2 5 8 + + = + + = + = + = + = + = + = = a b d a n EJERCICIO 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 1.1.1.1.1. ( ) ( )a b c d+ = + = = =1 2 3 4 3 3 4 9 4 5 2.2.2.2.2. ( )( ) ( )( )a b b a+ = + = =1 2 2 1 3 1 3 3.3.3.3.3. ( )( )b m c n a + = + = + = + = + = + = =4 2 1 2 3 2 3 4 1 4 1 2 9 2 3 4 3 2 7 3 4 7 2 4 7 8 2 15 2 72 2 1 2 4.4.4.4.4. ( )( ) ( )( )2 3 4 2 1 2 3 2 3 4 1 4 2 1 2 1 4 3 5 152 2 m n p b+ + = + + = + + = = 5.5.5.5.5. ( )( )( ) ( ) ( )4 8 6 4 1 2 8 1 4 1 2 6 2 3 4 2 2 3 0 02 2 2 2 m p a b n d+ + = + + = + =
  9. 9. 6.6.6.6.6. ( )( )( )( ) ( )( )( )c b d c b a m p = = =3 2 4 3 2 1 1 2 1 4 1 1 1 2 1 4 1 4 7.7.7.7.7. ( ) ( ) ( )b c d a m n x2 2 2 2 5 62 2 3 4 1 1 2 2 3 2 0 4 7 3 4 6 28 7 6 168 7 6 161 6 26+ + + = + + + = + = = = = 8.8.8.8.8. ( ) ( ) ( )2 6 4 2 1 2 0 6 2 3 4 4 6 4 9 4 16 6 13 64 78 64 142 2 2 2 2 2 mx b c d+ + = + + = + = = = 9.9.9.9.9. 8 9 16 8 1 2 9 2 3 16 1 4 2 1 4 18 3 4 2 4 6 2 2 3 2 2 6 3 8 3 2 2 3 m n p b a+ = + = + = + = + = + = = 10.10.10.10.10. ( ) ( ) ( )x m a d cb c a + + = + + = + = =0 1 2 1 4 3 1 2 1 64 3 1 2 62 312 3 1 11.11.11.11.11. ( )4 4 1 2 1 4 1 1 2 3 4 2 1 4 1 4 9 4 3 4 5 9 3 5 9 27 5 5 2 2 2 2 2 2 2 5 m p a a b c + + = + + = + + = = = = 12.12.12.12.12. ( )( )( ) ( )( )( ) 2 3 4 8 6 4 9 20 2 1 2 3 2 3 4 1 4 8 1 4 6 2 3 4 1 2 9 2 3 20 1 4 1 2 1 2 4 2 6 5 4 4 11 176 m n p p n m n p+ + + + = + + + + = + + + + = = 13.13.13.13.13. ( ) ( ) ( )c m n d m p b n p2 2 2 2 2 2 2 12 1 63 1 2 2 3 4 1 2 1 4 2 2 3 1 4 9 7 6 16 3 4 4 11 12 21 2 12 44 12 126 144 44 12 26 12 2 2 + + + + = + + + + = + = + = + = = = 14.14.14.14.14. c d a d m 2 2 2 2 1 2 2 3 4 1 2 4 1 2 9 16 1 2 2 1 2 25 1 2 2 1 2 5 1 2 5 2 2 + = + = + = = = = 15.15.15.15.15. ( )( ) ( )( ) ( )( ) ( )( ) 4 2 18 24 2 8 2 40 4 1 4 2 2 18 2 3 24 1 4 2 8 1 2 2 40 1 4 1 1 4 12 6 2 4 2 11 5 6 2 66 30 132 162 p b n p m p a+ + + + = + + + + = + + + = + = + = 16.16.16.16.16. a d b d b m p + + = + + = + + = + = + = = = = 5 2 1 4 2 4 2 5 2 1 2 1 4 2 4 2 2 5 2 1 4 1 16 6 2 2 5 8 1 16 3 2 5 8 1 16 3 2 13 1 16 3 2 208 624 2 312 2 2 2 2 17.17.17.17.17. ( ) ( )a b c b m n+ + = + + = = = = = = =2 2 2 2 4 6 2 38 1 2 3 8 2 1 2 2 3 3 25 1 2 2 3 3 5 2 6 15 2 6 90 2 6 88 6 14 14 18.18.18.18.18. ( ) ( ) a c n b c d p + + + = + + + = + = = 2 6 1 3 2 6 2 3 2 3 4 1 4 4 2 4 2 7 1 2 2 7 2 4 7
  10. 10. 19.19.19.19.19. ( ) ( ) ( ) ( ) 3 32 2 16 2 3 3 2 32 1 2 2 4 1 16 1 4 2 2 3 3 16 6 4 6 2 3 4 6 2 3 12 12 3 3 c b m d a p n = = = = = 20.20.20.20.20. ( ) ( ) 6 2 8 3 2 6 1 2 3 2 8 2 3 1 2 2 3 2 2 1 3 4 2 3 1 4 1 2 3 36 2 16 1 2 2 6 6 8 1 2 1 3 18 12 8 24 22 24 11 12 abc b mn b a cdnp abc + = + = + = + = + = = 21.21.21.21.21. ( )( ) ( )( )a b b a a b a b 2 2 2 2 2 2 2 2 2 33 2 3 1 2 2 1 3 1 2 2 1 3 2 5 3 3 3 8 5 3 72 5 216 3 221 3 73 + + + + = + + + + = + = + = + = = 22.22.22.22.22. b a b b c n m 2 2 2 2 2 2 1 2 1 1 1 1 1 1 2 1 1 1 2 1 2 1 3 1 2 3 1 1 2 4 3 2 5 6 3 2 2 4 5 4 7 2 21 4 49 4 70 4 35 2 17 + + + + + = + + + + + = + + + = + + = + = = = 23.23.23.23.23. ( )( ) ( )( ) 2 3 4 2 4 2 1 2 3 2 3 4 1 4 2 3 4 1 2 2 3 1 2 1 6 4 1 4 4 9 3 7 4 9 21 4 9 189 4 9 185 9 20 2 2 2 2 5 9 m n p c m n+ + = + + = + + = = = = = 24.24.24.24.24. b c ab m n b m 2 2 3 2 2 3 3 2 1 2 1 2 2 3 2 1 2 3 4 1 2 2 3 3 2 3 7 2 4 9 6 7 4 9 54 28 63 26 63 = = = = = = EJERCICIO 14 1.1.1.1.1. a b m+ + 2.2.2.2.2. m b x2 3 4 + + 3.3.3.3.3. a a+ +1 2; 4.4.4.4.4. x x 1 2; 5.5.5.5.5. y y y+ + +2 4 6; ; 6.6.6.6.6. ( )$ a x m+ + 7.7.7.7.7. m n 8.8.8.8.8. ( )bs x. 6 9.9.9.9.9. ( )x m Km 10.10.10.10.10. ( )$ x a m+ 11.11.11.11.11. ( )m a b c Km 12.12.12.12.12. ( )$ n 300 13.13.13.13.13. ( )365 x das 14.14.14.14.14. $ ; $15 ; $8a a ma 15.15.15.15.15. 2 3 2 a b c + + 16.16.16.16.16. a b m 2 17.17.17.17.17. 23 2 n m 18.18.18.18.18. x m2 2 19.19.19.19.19. ( ) ( )$ ; $3 6a b ax bm+ + 20.20.20.20.20. ( )( )a b x y+ + 21.21.21.21.21. ( )$ 8 48x + 22.22.22.22.22. ( )( )bs a x. +8 4 23.23.23.23.23. 75 x sucres 24.24.24.24.24. $ a m
  11. 11. 25.25.25.25.25. 3 000 1 . n colones 26.26.26.26.26. x a soles 3 27.27.27.27.27. m m 14 28.28.28.28.28. x a km h +1 EJERCICIO 15 29.29.29.29.29. $ a b m + 2 30.30.30.30.30. x x x hab+ + 2 2 . 31.31.31.31.31. 1000 3 2 4 . + + a a a sucres 1.1.1.1.1. m n m n, + 2.2.2.2.2. ( )m n m n m n, + = 3.3.3.3.3. +3 4 3 4a b a b, 4.4.4.4.4. 5 6 6 5b a a b, + 5.5.5.5.5. ( )7 6 7 6 7 6 1, + = = 6.6.6.6.6. + =6 9 6 9 3, 7.7.7.7.7. +2 3 2 3x y x y, 8.8.8.8.8. ( )5 5 5mn m mn m mn m, + = 9.9.9.9.9. 5 7 5 7 12a a a a a, + = 10.10.10.10.10. ( ) + = = 8 5 8 5 8 5 13x x x x x x x, 11.11.11.11.11. + = 11 8 11 8 3m m m m m, 12.12.12.12.12. 9 15 9 15 6ab ab ab ab ab, = 13.13.13.13.13. = xy xy xy xy xy, 9 9 10 14.14.14.14.14. ( )mn mn mn mn mn mn mn, + = = 11 11 11 10 15.15.15.15.15. 1 2 2 3 1 2 2 3 a b a b, 16.16.16.16.16. 3 5 3 4 3 5 3 4 b c b c, + 17.17.17.17.17. 1 3 2 3 1 3 2 3 3 3 b b b b b b, + = = 18.18.18.18.18. = = 1 2 1 2 1 2 1 2 2 2 xy xy xy xy xy xy, 19.19.19.19.19. = = 3 5 2 5 3 5 2 5 5 5 abc abc abc abc abc abc , 20.20.20.20.20. + = + = 4 3 8 4 3 8 32 3 8 29 8 2 2 2 2 2 2 x y x y x y x y x y x y , 21.21.21.21.21. 3 8 3 4 3 8 3 4 3 6 8 3 8 mn mn mn mn mn mn , = = 22.22.22.22.22. a b c a b c, , + + 23.23.23.23.23. a b c a b c, , + 24.24.24.24.24. a b c a b c, , +2 2 25.25.25.25.25. 3 2 4 3 2 4m n p m n p, , + 26.26.26.26.26. a ab b a ab b2 2 2 2 7 5 7 5, , 27.27.27.27.27. x xy y x xy y2 2 2 2 3 4 3 4, , 28.28.28.28.28. x x y x x y3 2 3 2 6 6, , + 29.29.29.29.29. 2 3 2 3 5a b a a a b a b, , + = 30.30.30.30.30. + = m n n m n n m n, ,8 4 8 4 4 31.31.31.31.31. + = 7 8 7 8a a b a a b a b, , 32.32.32.32.32. 1 2 2 3 3 4 1 2 3 4 2 3 4 6 8 2 3 1 4 2 3 x y x x x y x y x y , , + = + = + 33.33.33.33.33. = = 3 5 2 3 3 5 2 3 3 5 5 2 3 8 5 2 3 m m mn m m mn m mn m mn , , 34.34.34.34.34. + + = + + 7 5 3 7 5 3 8 5 3 2 2 2 2 2 2 2 2 a ab b a a a ab b a ab b , , ,
  12. 12. 35.35.35.35.35. + = 7 17 5 4 10 92 2 2 mn mn m m mn m 36.36.36.36.36. x x x y x y x x y3 3 2 2 3 2 7 8 4 5 6 4 5 + + = + 37.37.37.37.37. 5 9 6 7 4 3 72 2 2 2 2 x x xy xy y x xy y + + = + + 38.38.38.38.38. + = 8 5 11 7 9 6 7 2 2 2 2 3 2 2 3 a b a b ab ab b a b ab b 39.39.39.39.39. m m n m n mn n m m n mn n 3 2 2 2 3 3 2 2 3 8 7 7 7 + + = + 40.40.40.40.40. 1 2 1 4 2 3 1 5 6 2 1 4 10 3 15 6 1 4 13 15 6 a a b b a b a b + + = + + = + 41.41.41.41.41. a a b b c c b + + =3 4 8 8 42.42.42.42.42. m m m m n m n mn m m n mn 3 3 3 2 2 2 3 2 2 5 5 4 4 7 8 7 + = 43.43.43.43.43. 9 11 6 4 6 8 17 2x x y y z z x y z + = 44.44.44.44.44. 6 9 5 7 8 11 15 5 15 11 2 2 2 2 2 2 a a ab b b a ab b + = 45.45.45.45.45. + + = x y x y xy xy y xy y 2 2 2 2 3 3 4 3 4 5 7 4 8 2 4 8 46.46.46.46.46. 3 1 2 1 2 4 6 6 1 2 1 2 2 2 5 2 1 2 2 a a b b a b a b + + = + + = + 47.47.47.47.47. 1 2 3 4 2 3 1 3 5 6 5 6 2 3 4 2 1 3 5 4 1 3 2 2 2 2 2 2 x x xy xy y y x xy x xy + + + = + + = + 48.48.48.48.48. 5 5 6 5 8 81 1 1 2 2 ax ax a a a a ax x x x x + + + =+ + + + + 49.49.49.49.49. 3 4 2 3 1 3 1 3 5 3 4 4 3 3 1 15 3 7 4 16 3 2 2 2 2 2 2 2 2 x x xy xy y y x xy y x xy y + + + = + + + = + 50.50.50.50.50. 3 4 1 4 1 2 1 2 5 6 3 1 4 4 3 3 5 6 3 2 1 6 2 2 2 2 2 2 2 2 2 2 a b a b a b ab ab ab a b ab a b ab + + + = + + + = + EJERCICIO 16 1.1.1.1.1. 3 2 2 3 5 5 a b c a b c a b + + + + 2.2.2.2.2. 7 4 5 7 4 6 a b c a b c c + + 3.3.3.3.3. m n p m n p + + 0 4.4.4.4.4. 9 3 5 4 5 4 9 3 x y x y x y x + + + 5.5.5.5.5. a b c a b c a b c b + + + 2 2 2 3 3 2 6.6.6.6.6. p q r p q r p q r r + + + + 2 6 3 5 8 4 7.7.7.7.7. + + + 7 4 6 10 20 8 5 24 2 2 x y z x y z x y z x 8.8.8.8.8. + + + 2 3 6 3 8 8 5 10 4 4 8 m n m n m n m n 9.9.9.9.9. + + 5 2 3 7 3 5 8 5 3 6 a b c a b c a b c a c 10.10.10.10.10. ab bc cd ab bc cd ab bc cd ab + + + + 8 3 3 5 2 2 2 11.11.11.11.11. ax ay az ax ay az ax ay az ay az + + + 5 7 6 4 9 8 12.12.12.12.12. 5 7 8 4 6 3 8 9 2 23 x y x y x y x + + + + + 13.13.13.13.13. + + am mn s am mn s am mn s am mn 6 4 5 6 3 5 2 4 14.14.14.14.14. 2 3 6 4 8 9 4 a b b c a c a b c + + + + 15.15.15.15.15. 6 3 4 5 5 5 7 m n n p m p m n + 16.16.16.16.16. 2 3 5 4 8 6 7 9 10 3 12 7 a b c a c a b c + + + + + + 17.17.17.17.17. 2 3 5 9 6 4 3 5 8 5 x y z x y x z + + +
  13. 13. 21.21.21.21.21. 5 3 7 8 5 9 11 5 16 14 7 a a a a a a a a a a a x m n x m n x m n x m + + + + 22.22.22.22.22. 6 7 5 4 7 5 3 12 5 11 6 1 2 3 1 2 3 1 2 3 1 2 3 m m m m m m m m m m m m a a a a a a a a a a a a + + + + + + + + + + + + + + + 23.23.23.23.23. 8 3 4 2 3 4 5 3 4 9 2 3 2 x y z u x y z u x y z u x y z u y z u + + + + + + + + + + 7.7.7.7.7. m n n mn m n m mn 2 2 2 2 2 2 4 3 5 5 4 3 + + 8.8.8.8.8. 3 4 5 4 6 3 1 3 2 3 2 x x x x x x + + + 9.9.9.9.9. x xy y x xy y x xy y x xy y 2 2 2 2 2 2 2 2 3 3 2 3 3 2 + + + + 10.10.10.10.10. a ab b a ab b a ab b b 2 2 2 2 2 2 2 3 5 2 8 + + + 11.11.11.11.11. + + + + 7 5 6 4 8 9 7 14 4 6 1 2 2 2 2 x x x x x x x x 18.18.18.18.18. 8 3 5 7 4 19 3 a b c a b c a b c a b c a c + + + + 19.19.19.19.19. 7 2 4 9 6 5 3 6 8 3 5 15 7 3 10 x y y z y z x y x y z + + + + + 20.20.20.20.20. + + + + + + m n p m n m p m n m n p 2 5 6 3 4 5 2 8 3 2 9 24.24.24.24.24. a b c d a b c d a b c d a b c d a c + + + + + + + 2 3 2 3 3 4 3 2 25.25.25.25.25. 5 3 4 2 2 3 2 4 3 3 6 2 ab bc cd bc cd de ab bc de ab bc cd ab + + + + + 26.26.26.26.26. a b b c c d a c c d a d a d a + + 2 12.12.12.12.12. a a a a a a a a a 3 3 2 2 3 2 4 5 2 6 7 4 2 11 15 + + + + + 13.13.13.13.13. + + + + x x x x x x x x 2 3 2 3 2 6 7 5 8 5 8 9 6 14.14.14.14.14. a b a b ab a ab b a a b ab b 3 3 2 2 3 2 3 3 2 2 3 5 4 7 2 5 11 2 + + 15.15.15.15.15. x xy y x x y y x xy y x x y xy y 3 2 3 3 2 3 3 2 3 3 2 2 3 5 2 4 5 4 5 3 5 + + 16.16.16.16.16. + + + + + + + 7 4 6 7 5 6 8 2 3 3 2 3 3 2 3 2 3 m n n m mn n m m n n mn n 1.1.1.1.1. x x x x x x 2 2 2 4 5 2 + 2.2.2.2.2. a ab ab b a ab b 2 2 2 2 2 + + + 3.3.3.3.3. x x x x x x 3 2 3 2 2 4 2 4 + + + + 4.4.4.4.4. a a a a a a a a 4 2 3 4 3 2 3 4 3 4 + + + + 5.5.5.5.5. + + + + x x x x x x 2 3 3 2 3 6 3 6 6.6.6.6.6. x x x x x x 2 2 2 4 7 6 3 5 4 11 1 + + EJERCICIO 17
  14. 14. 17.17.17.17.17. x x x x x x x x x x x 4 2 3 2 2 4 3 2 4 5 7 4 6 2 3 11 + + + + + + + + 18.18.18.18.18. + + + + + + + + + a a a a a a a a a a a 6 4 5 3 3 2 6 5 4 3 2 6 3 8 14 2 19.19.19.19.19. x x x x x x x x x x x 5 4 2 3 5 4 3 2 9 3 7 6 3 4 5 3 3 7 3 2 + + + + + + 20.20.20.20.20. a a a a a a a a 3 2 2 2 3 5 4 7 8 6 5 1 + + + + + + 21.21.21.21.21. x x y x y xy xy y x y x x y x y xy y 4 2 2 3 3 3 4 2 2 4 3 2 2 3 4 5 6 4 4 6 5 5 2 6 + + + + 22.22.22.22.22. + + + + + + + x xy x xy y x xy y x xy y x xy y 2 2 2 2 2 2 2 2 2 4 7 6 5 6 4 7 7 23.23.23.23.23. a ax x ax x a x a x a x a ax x a x 3 2 3 2 3 2 3 3 2 3 2 3 3 3 8 6 5 3 5 14 5 2 + + + 24.24.24.24.24. + + + + + + 8 6 5 4 4 3 7 4 6 3 3 6 6 2 2 3 3 2 3 3 2 2 2 2 3 2 2 a m am m a am m a a m am a m am a a m am 25.25.25.25.25. x x y xy x y x y y x y xy y x xy y x x y x y x y 5 3 2 4 4 2 3 5 3 2 4 5 5 4 5 5 4 3 2 2 3 2 3 3 4 5 2 2 2 2 3 + + + + + + + + 26.26.26.26.26. a a a a a a a a a a a a a 5 6 2 4 3 2 5 2 6 4 3 6 3 8 5 4 6 5 4 + + + + + + + + + + + + 27.27.27.27.27. a b a b a b ab a a b a b a b a b b a ab b 4 4 3 2 2 3 4 3 2 2 3 2 2 4 4 3 4 3 5 4 4 3 3 2 4 + + + 28.28.28.28.28. m n m n n m n mn m n mn m n m n mn 3 3 2 3 2 2 3 3 2 3 3 2 2 6 4 5 6 2 2 11 + + + + + 29.29.29.29.29. a a a a a a a a a a a a x x x x x x x x x x x x + + + + + + + 3 5 6 7 13 6 3 2 1 3 3 4 1 3 1 2 4 30.30.30.30.30. + + + + + + + + + + + + + + + + + + + + + a a a a a a a a a a a a a a a a a a a a x x x x x x x x x x x x x x x x x x x x 1 2 3 1 2 3 2 1 2 2 1 3 2 3 2 1 3 4 5 2 3 3 2
  15. 15. EJERCICIO 18 1.1.1.1.1. 1 2 1 3 1 2 1 4 1 2 2 3 6 1 4 1 2 5 6 1 4 2 2 2 2 2 2 x xy xy y x xy y x xy y + + + + + + = + + 2.2.2.2.2. a ab ab b ab b a ab b a b 2 2 2 2 2 2 2 1 2 1 4 1 2 1 4 1 5 2 1 1 4 5 2 10 3 10 + + + + = + 3.3.3.3.3. x xy xy y xy y x xy y x xy y x xy y 2 2 2 2 2 2 2 2 2 2 3 1 6 5 6 2 3 4 1 5 6 3 2 3 2 6 5 3 1 3 5 3 + + + + + + = + = + 4.4.4.4.4. 3 4 1 2 2 5 1 6 1 10 1 3 3 4 4 1 10 3 1 2 6 3 4 3 10 2 2 2 2 2 2 2 x y xy y xy y x xy y x xy + + + + + + + + = 5.5.5.5.5. 2 3 1 5 1 2 5 6 1 10 1 6 1 12 1 20 1 3 8 10 1 12 4 2 1 20 3 1 2 6 17 12 3 20 4 6 17 12 3 20 2 3 2 2 2 2 2 2 2 2 2 2 2 2 a ab b a ab b a ab b a ab b a ab b a ab b + + + + + + + + = + = + 6.6.6.6.6. 5 6 2 3 3 4 1 6 1 8 1 2 1 3 1 4 5 6 5 1 2 6 16 3 6 24 9 6 10 12 2 6 7 24 13 12 1 3 13 12 7 24 2 2 2 2 2 2 2 2 2 2 2 2 x y xy x y xy x y xy x y xy x y xy x xy y + + + + + + + + + = + + 7.7.7.7.7. a ab b ab b a b a b a b a ab b a b a ab b a b a a b ab b 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3 2 3 2 2 3 1 2 3 8 2 5 6 1 4 3 5 1 2 4 1 4 4 3 8 5 10 3 5 5 3 6 5 4 7 8 8 5 2 6 5 4 1 3 7 8 8 5 + + + + + + = + + 8.8.8.8.8. x x x x x x x x x x x x x x x x x x x 4 2 3 4 3 4 2 3 4 2 3 4 3 2 5 3 2 3 3 8 3 5 5 6 3 4 5 3 5 2 4 5 6 3 6 8 2 5 2 9 6 9 8 2 5 3 2 9 8 2 + + + + + + + = + + + + 9.9.9.9.9. 2 3 1 4 2 5 1 8 3 5 1 6 1 2 2 3 3 2 1 8 2 3 5 5 1 3 6 5 3 1 8 6 5 2 6 5 3 1 3 1 8 6 5 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3 2 3 2 2 3 m mn n mn n m n m n m n m mn n m n m mn n m n m m n mn n + + + + + + + + = 10.10.10.10.10. x x y y x x y y xy x y y x y x x y y xy x y x x y y xy x y x x y x y xy y 4 2 2 4 4 2 2 4 3 2 2 4 3 4 2 2 4 3 3 4 2 2 4 3 3 4 3 2 2 3 4 2 2 7 5 6 3 8 1 14 1 6 1 4 1 7 5 6 6 5 6 16 3 2 8 4 1 2 14 1 6 5 6 1 6 17 8 5 14 1 6 5 6 1 6 5 6 17 8 1 6 5 14 + + + + + + + + = + + + +
  16. 16. 11.11.11.11.11. x x x x x x x x x x x x x x x x x x x x x x x x x x 5 3 5 2 3 2 4 3 5 3 2 4 5 3 2 4 5 4 3 2 2 3 4 5 3 1 10 3 8 1 6 1 4 2 3 1 12 3 5 4 2 8 2 1 12 8 1 6 10 3 2 8 2 3 4 2 7 12 13 10 1 8 2 3 4 2 2 3 7 12 1 8 13 10 4 + + + + + + + + + = + + + + 12.12.12.12.12. 2 9 5 6 1 3 3 7 7 8 1 9 2 3 1 2 1 4 2 6 9 6 7 14 20 21 6 24 3 1 9 4 9 1 14 7 24 4 9 3 2 3 2 2 3 3 2 2 3 2 2 3 3 2 2 3 a ax x a x ax x a a x ax a a x ax x a a x ax x + + + + + + = + 13.13.13.13.13. a a a a a a a a a a a a a a a a a a a a a a a a a a a 6 4 2 5 3 4 2 6 4 2 5 3 6 4 2 5 3 6 5 4 3 2 3 5 3 8 1 2 3 7 5 8 6 3 8 6 7 3 7 8 5 8 3 5 3 8 4 3 8 10 7 3 8 3 5 3 8 7 8 3 5 10 7 3 8 3 8 7 8 + + + + + + + = + + + + 14.14.14.14.14. x y x y xy y y x y x y x y y x y x x y xy y x y x y x x y xy y x y x y x x y x y x y xy y 5 5 3 2 4 5 5 4 2 3 3 2 5 4 5 3 2 4 5 4 2 3 5 3 2 4 5 4 2 3 5 4 3 2 2 3 4 5 1 10 3 4 1 6 1 9 3 5 5 6 2 5 1 3 2 1 4 10 3 4 18 3 2 6 18 3 10 5 5 6 3 10 3 4 29 18 13 5 5 6 13 5 3 10 5 6 3 4 29 18 + + + + + + + = + + EJERCICIO 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 1.1.1.1.1. 4 5 3 6 8 2 8 2 4 8 8 8 0 x y x y x y y + + = = 2.2.2.2.2. x x x x x x x x 2 2 2 2 2 5 8 10 30 6 5 50 6 10 72 6 5 10 5 72 6 25 50 72 150 50 72 172 + + + + + = + = + = 3.3.3.3.3. x y x x y x x y xy x y x y x y xy x x y x y xy y 4 4 4 2 2 4 3 3 4 4 2 2 3 3 4 3 2 2 3 4 4 3 2 2 3 4 2 5 8 4 7 10 5 8 7 10 7 5 10 8 5 7 5 4 5 5 4 10 5 4 4 8 625 28 125 5 25 16 50 64 256 8 625 3 500 2 000 3 200 256 8 3 811 + + + + + + = + + = + + = + + =. . . . 4.4.4.4.4. 3 5 6 6 20 8 12 20 12 9 45 2 9 2 3 45 1 5 2 18 3 45 5 2 6 9 2 1 m n m n m n m n + + + + = + = + =
  17. 17. 5.5.5.5.5. nx cn ab nx cn ab nx ab nx cn ab + = = = 8 2 5 10 3 5 10 1 5 5 10 1 5 3 2 3 5 50 5 10 5 6 3 5 10 2 18 5 15 6.6.6.6.6. a b a b ab b a ab a b b a ab b 3 3 2 2 3 3 2 2 3 3 2 3 3 2 3 3 8 5 6 8 3 2 4 2 2 8 4 2 2 2 3 2 3 8 4 8 4 9 2 27 8 32 36 54 8 42 + + + + + + + + = + + = + + = 7.7.7.7.7. 27 125 9 25 14 8 10 11 27 22 125 8 27 2 3 2 3 1 5 22 2 3 1 5 125 1 5 8 27 8 27 4 9 1 5 44 3 1 25 125 1 125 8 8 4 45 44 75 1 8 4 45 44 75 1 20 132 3 3 2 2 2 2 2 3 2 2 3 3 2 2 3 m n m n mn mn m n mn m m n mn n + + + + + + + + + + = + + + = + + + = + + = + + 225 225 377 225 1152 225= = 8.8.8.8.8. x y m x y m y m x y m a b x a b x b x a b x + + + + + = + = 1 2 4 1 2 4 2 4 1 2 4 2 1 3 2 5 4 2 2 2 3 2 3 2 3 3 5 2 4 3 2 3 15 8 2 21 9.9.9.9.9. n m n m n m m b x b x b x x + + + = = 1 3 1 3 1 3 3 5 3 2 8 5 3 10 4 5 18 2 3 2 3 4 9 10.10.10.10.10. x y xy x y x x y xy x y xy y x y xy x y x x y xy y 3 3 3 4 2 2 3 2 2 3 4 3 3 4 4 4 3 3 4 4 3 3 4 5 5 6 6 2 3 1 6 4 2 6 4 2 5 6 5 4 4 5 4 4 2 625 24 125 20 64 256 2 625 3 000 1280 256 2 2 091 + + + + + + + + + + = + + = + + = + + =. . . 11.11.11.11.11. 3 4 2 3 1 3 1 9 1 6 1 3 3 4 2 1 6 6 1 3 9 3 4 1 6 4 9 3 4 2 1 6 2 3 4 9 3 3 4 4 1 6 6 4 9 9 3 1 4 6 2 2 2 2 2 2 2 2 2 2 a b ab b ab b a ab b a ab b + + + + + + = + + = + = + = 12.12.12.12.12. 9 17 25 34 1 4 7 34 5 17 1 2 15 7 34 1 4 30 3 18 7 7 34 25 10 34 1 2 1 12 4 45 18 34 35 34 12 4 45 9 17 45 35 34 3 9 17 2 3 45 2 3 1 5 35 34 1 5 3 9 17 4 9 90 15 35 34 1 25 3 2 2 2 2 2 2 2 2 2 2 2 2 2 m n m n mn m mn m n mn m n mn m mn n + + + + + + + + + + = + + + + = + + = + + = + + = + = + = = 4 17 6 35 850 3 4 17 7 170 3 40 7 510 170 463 170 2 123 170
  18. 18. 14.14.14.14.14. 0 2 0 4 0 5 0 6 0 3 0 8 0 4 0 8 6 0 2 15 0 9 01 01 6 01 01 6 01 2 3 2 3 01 3 6 0 3 4 2 9 01 27 6 12 18 2 7 6 25 5 3 2 2 2 2 3 3 2 3 2 3 2 2 3 2 2 3 2 2 3 , , , , , , , , , , , , , , , , , , , , , , a ab a b ab a b b a a b a a b b ab a b b a b ab b + + + + + + + + + + = + + + = + + + = + + + = 13.13.13.13.13. 1 2 3 5 2 3 4 1 10 6 1 4 1 25 4 1 8 2 3 5 4 6 2 1 8 30 5 2 100 50 10 30 20 3 5 7 8 67 50 43 5 7 8 3 2 3 67 50 10 1 5 43 5 7 9 2 24 67 50 43 5 126 24 670 250 43 5 21 4 67 25 43 5 525 268 860 100 1 653 100 16 2 2 2 2 2 2 2 53 100 b m cn b m cn b m cn b m cn b m cn b m cn + + + + + + + + + + + + + + + = + + = + + = + + = + + = + + = = . EJERCICIO 20 1.1.1.1.1. = 8 5 13 2.2.2.2.2. = 7 4 11 3.3.3.3.3. 8 11 3 = 4.4.4.4.4. ( ) = + =8 11 8 11 3 5.5.5.5.5. ( ) = + =1 9 1 9 8 6.6.6.6.6. 2 3 2 3a b a b = 7.7.7.7.7. 3 2 3 2b b = 8.8.8.8.8. 4 6 4 6x b x b = 9.9.9.9.9. = 5 6 5 6a b a b 10.10.10.10.10. ( ) = +8 3 8 3x x 11.11.11.11.11. = 9 5 9 52 2 2 2 a b a b 12.12.12.12.12. ( ) = +7 5 7 5xy yz xy yz 13.13.13.13.13. 3 4a a a = 14.14.14.14.14. 11 25 142 2 2 m m m = 15.15.15.15.15. ( ) = + = 6 6 52 2 2 2 2 x y x y x y x y x y 16.16.16.16.16. ( )11 7 11 7 183 2 3 2 3 2 3 2 3 2 a m a m a m a m a m = + = 17.17.17.17.17. ( ) = + =8 8 8 8 02 2 2 2 ab ab ab ab 18.18.18.18.18. ( )31 46 31 46 772 2 2 2 2 x y x y x y x y x y = + = 19.19.19.19.19. ( ) = + =84 84 84 84 02 2 2 2 a b a b a b a b 20.20.20.20.20. 3 5 3 51 2 1 2 a b a bx x x x+ + + + = 21.21.21.21.21. = + + 8 11 8 112 2 x xa a 22.22.22.22.22. ( )6 5 6 5 11a a a a an n n n n = + = 23.23.23.23.23. ( ) = + = 45 60 45 60 151 1 1 1 1 a a a a ax x x x x 24.24.24.24.24. ( )54 86 54 86 1401 1 1 1 1 b b b b bn n n n n = + = 25.25.25.25.25. ( ) = + =35 60 35 60 25m m m m ma a a a a 26.26.26.26.26. 5 1 2 5 1 2 10 1 2 11 2 5 1 2 = + = + = = 27.27.27.27.27. = = 2 3 3 4 8 9 12 17 12 28.28.28.28.28. 1 3 2 3 1 3 2 3 3 3 2 2 2 2 2 2 x x x x x x = + = = 29.29.29.29.29. 4 5 5 6 24 25 30 49 30 3 3 3 3 x y x y x y x y = + = 30.30.30.30.30. = + = 1 8 3 4 1 6 8 5 8 2 2 2 2 ab ab ab ab 31.31.31.31.31. = 2 3 5 32.32.32.32.32. ( )7 1 7 1 8 = + = 33.33.33.33.33. ( ) = + = 8 5 8 5 3 34.34.34.34.34. ( )5 4 5 4 9 = + = 35.35.35.35.35. ( ) = + =7 7 7 7 0 36.36.36.36.36. ( )2 5 2 5a a = +
  19. 19. 37.37.37.37.37. = 3 3x b x b 38.38.38.38.38. = 2 5 2 5n m n m 39.39.39.39.39. ( )3 6 6 3b a a b = + 40.40.40.40.40. ( )8 5 5 83 3 b a a b = + 41.41.41.41.41. ( ) = +7 9 7 9a a 42.42.42.42.42. ( )25 25 25 25ab ab = + 43.43.43.43.43. ( )3 3 4a a a a a = + = 44.44.44.44.44. ( ) = + = 4 3 4 3b b b b b 45.45.45.45.45. ( )54 11 54 11 653 3 3 3 3 x x x x x = + = 46.46.46.46.46. 78 14 642 2 2 a b a b a b = 47.47.47.47.47. ( ) = + = 54 43 54 43 112 2 2 2 2 a y a y a y a y a y 48.48.48.48.48. = ab ab ab9 10 49.49.49.49.49. ( ) = + =31 31 31 31 02 2 2 2 x y x y x y x y 50.50.50.50.50. = 3 4a a ax x x 51.51.51.51.51. ( )311 7 311 7 3181 1 1 1 1 a a a a ax x x x x+ + + + + = + = 52.52.52.52.52. 105 9 96m m mx x x = 53.53.53.53.53. = 31 18 491 1 1 a a ax x x 54.54.54.54.54. ( ) = + = 236 19 236 19 217m m m m ma a a a a 55.55.55.55.55. = + + + 85 54 1392 2 2 a a ax x x 56.56.56.56.56. ( )1 4 6 1 4 6 = +a a 57.57.57.57.57. ( ) = + = 2 3 5 2 15 3 13 3 58.58.58.58.58. = = = 7 10 3 8 56 30 80 86 80 43 40 3 3 3 3 3 m m m m m 59.59.59.59.59. 5 6 11 12 10 11 12 21 12 7 4 2 2 2 2 2 2 2 2 2 2 a b a b a b a b a b = + = = 60.60.60.60.60. = = = 1 9 45 1 405 9 406 9 45 3 2 3 2 3 2 3 2 1 9 3 2 a b a b a b a b a b EJERCICIO 21 1.1.1.1.1. a b a b b + + 2 2.2.2.2.2. 2 3 2 3 5 x y x y x y 3.3.3.3.3. 8 3 4 11 4 a b a a b + + 4.4.4.4.4. x x x x x 2 2 3 5 6 2 6 + 5.5.5.5.5. a a b a b ab a a b ab 3 2 2 2 3 2 2 7 9 8 9 6.6.6.6.6. x y z x y z + + 0 7.7.7.7.7. x y z x y z x y z + + + 2 2 2 8.8.8.8.8. x y xy x y xy x y xy x xy y 2 2 2 2 2 2 2 2 3 3 4 2 2 2 2 + + + + + + + 9.9.9.9.9. x x x x x x x 3 2 2 3 2 6 5 6 4 6 4 + + + 10.10.10.10.10. y y y y y y y y y y y y y 2 3 2 4 2 4 3 4 3 2 6 8 3 2 6 4 2 6 6 8 2 6 4 6 8 + + + + 11.11.11.11.11. a ab a a a b a ab a a b a a b ab a 3 2 2 3 2 2 3 2 2 6 9 8 15 5 6 17 15 5 15 6 17 5 + + + + 12.12.12.12.12. x xy y y x y x y x xy y x y x y x x y x y xy y 4 3 4 4 3 2 2 4 3 4 3 2 2 4 3 2 2 3 4 9 11 20 8 6 9 31 8 6 8 6 9 31 + + + + + + + + + 13.13.13.13.13. a b c d a b c d a b + + + + +2 2 14.14.14.14.14. ab ac cd de ab ac cd de ab ac cd de + + + + + 2 3 5 8 4 5 5 7 6 2 10 15.15.15.15.15. x x x x x x x x x 3 2 3 2 3 2 6 9 19 6 11 21 43 5 17 30 24 + + + + +
  20. 20. 16.16.16.16.16. y y y y y y y y y y y y y y y y y 5 3 2 3 2 4 5 3 2 4 5 4 3 2 9 6 31 31 8 11 19 40 14 11 19 31 11 40 14 19 31 + + + + + + + + + + 17.17.17.17.17. 5 9 6 8 5 21 14 18 9 27 22 18 27 22 9 18 3 3 2 2 3 2 2 3 2 2 2 2 3 m n m n mn m m n mn n m n mn m n mn n + + + + + + 18.18.18.18.18. 4 19 6 25 51 32 29 32 38 29 38 32 3 3 4 2 2 3 3 2 2 4 3 3 4 2 2 4 4 3 2 2 3 4 x y xy y x y x y xy x y x x y xy y x y x x x y x y xy y + + + + + + + + + 19.19.19.19.19. m m n m n m n m n mn m m n m n m n mn m m n m n m n mn 6 4 2 2 4 2 4 3 3 5 6 4 2 2 4 3 3 5 6 4 2 3 3 2 4 5 9 19 30 61 13 16 21 80 13 16 13 21 16 80 + + + + + + + + + + + + + 20.20.20.20.20. + + + + + + + + a b a b ab a a b a b b a a b a b a b a b ab b 5 3 3 5 6 4 2 2 4 6 6 5 4 2 3 3 2 4 5 6 6 18 42 8 11 11 9 8 11 6 11 18 9 42 21.21.21.21.21. 1 3 6 24 30 8 15 25 29 7 12 6 6 7 29 12 25 2 4 3 5 2 4 6 2 4 3 5 6 6 5 4 3 2 + + + + + + + + x x x x x x x x x x x x x x x x x x x x x 22.22.22.22.22. + + + + + + + + + + + + 6 8 23 80 18 51 21 80 9 6 8 28 101 98 9 8 28 101 6 9 98 2 3 5 4 3 2 4 3 2 5 4 2 3 5 4 3 2 5 4 5 4 3 2 2 3 4 5 x y x x y x y x y x y y xy x y x x y x y y xy x x y x y x y xy y 23.23.23.23.23. m m n m n mn m n m n mn n m m n m n m n m n mn n 6 4 2 2 4 5 5 3 3 5 6 6 5 4 2 3 3 2 4 5 6 8 21 6 8 23 14 24 8 14 23 8 14 21 18 8 22 + + + + + + + + 24.24.24.24.24. x x x x x x x x x x x x x x x x x x x x x x 7 5 2 6 4 3 7 5 2 6 4 3 7 6 5 4 3 2 8 16 23 15 51 18 8 25 30 59 16 23 3 8 25 30 8 16 25 30 23 59 3 + + + + + + + + + + + + 25.25.25.25.25. 9 15 31 14 15 3 25 53 9 9 31 4 25 53 9 14 9 25 53 31 9 4 14 6 4 2 2 4 6 4 2 6 5 3 3 5 6 2 4 6 5 3 3 5 6 5 3 3 2 4 5 6 a a b a b b a b b a b a b ab a a b b a b a b ab a a b a b a b ab b + + + + + + + + + + 26.26.26.26.26. a a a a a a a a x x x x x x x x + + + + + + + + + 1 2 1 2 1 5 6 4 7
  21. 21. 27.27.27.27.27. m m m m m m m m m m m m m m m m m a a a a a a a a a a a a a a a a a + + + + + 1 2 2 1 3 1 2 1 3 1 1 2 3 3 4 5 3 8 5 2 3 8 3 5 2 8 28.28.28.28.28. a a a a a a a a a a a a a a a a a a a a m m m m m m m m m m m m m m m m m m m m + + + + + + + + + + + + + + + + + + + + + + + + + + 4 2 1 2 1 3 1 4 2 1 3 1 4 3 2 1 1 7 8 6 14 8 5 11 7 8 14 5 11 5 7 11 8 14 29.29.29.29.29. x x x x x x x x x x x x x x x x x x x x a a a a a a a a a a a a a a a a a a a a + + + + + + + + + + + + + 2 1 2 1 1 3 2 1 2 1 3 2 1 1 2 3 7 9 25 19 45 11 60 26 36 25 11 60 11 26 36 25 60 30.30.30.30.30. m m m m m m m m m m m m m m m m m m m m m n n n n n n n n n n n n n n n n n n n n n + + + + + + 1 2 3 5 2 3 5 4 1 2 3 5 4 1 2 3 4 5 6 8 19 5 6 9 8 11 2 28 8 8 11 2 28 EJERCICIO 22 1.1.1.1.1. b a b a b a a b +2 2 2 2 2.2.2.2.2. 2 3 4 x y x y x y + + + 3.3.3.3.3. + + + 7 5 5 2 5 2 5 a a b a b a b 4.4.4.4.4. + + + + + + x x x x x x x 2 2 2 2 6 5 2 6 5 2 5 6 5.5.5.5.5. x y xy xy x x y xy x x x y xy 2 2 2 3 2 2 3 3 2 2 5 6 6 + + + + + 6.6.6.6.6. 7 5 6 8 5 8 8 5 2 2 2 3 2 2 3 3 2 2 a b ab a b a a b ab a a a b ab + + + + + + 7.7.7.7.7. + + + a b c a b c a b c 2 3 2 2 3 5 8.8.8.8.8. + + + + + + 3 4 5 2 3 4 3 2 4 n m p n m p n m p m n p 9.9.9.9.9. x y z x y z x y z + + + 3 6 2 2 5 10.10.10.10.10. + + + + + 5 8 6 3 7 2 2 7 2 2 2 2 2 2 2 2 b ab a b ab a b ab a a ab b 11.11.11.11.11. + + + + 5 6 3 6 9 2 2 2 2 2 m n mn m n mn m mn 12.12.12.12.12. + + + + + + 8 5 4 6 8 6 10 8 6 10 2 3 2 3 3 2 x x x x x x x x x x
  22. 22. 13.13.13.13.13. 14 8 16 14 9 8 7 8 7 2 2 3 3 3 m n m m n m m n + + + 14.14.14.14.14. 8 5 6 6 7 6 ab bc cd ab bc cd ab bc + + + + 15.15.15.15.15. a a b b a b b ab a a b ab 3 2 3 2 3 2 3 2 2 9 25 8 34 8 + + + 16.16.16.16.16. 6 8 6 6 4 6 8 7 6 4 3 2 2 2 3 3 2 2 3 x x y xy xy y x x y xy y + + 17.17.17.17.17. m n c m n c d n c d c d n 2 2 9 11 14 7 8 16 19 14 19 16 14 + + + + + + 18.18.18.18.18. 5 9 40 6 7 8 5 5 2 8 45 5 4 3 3 4 3 2 2 3 4 4 3 2 2 3 4 a a b ab b a b a b ab b a a b a b ab b + + + + + + 19.19.19.19.19. x x x x x x x x x x x x x x x x 5 4 2 2 3 5 4 2 3 5 4 3 2 8 25 15 6 7 6 9 8 19 22 6 9 8 6 19 9 22 + + + + + + + + + + 20.20.20.20.20. + + + + + + 3 8 19 18 6 25 9 8 44 18 8 9 44 18 4 3 2 5 4 5 5 2 3 4 3 2 5 5 2 3 5 3 2 2 3 4 5 xy x y y xy y x x y xy x y y x x y x x y x y xy y 21.21.21.21.21. x x x x x x x x x x x x x x x x x x 3 4 2 3 2 5 3 4 2 5 5 4 3 2 6 8 9 15 25 18 46 25 11 24 6 26 37 10 11 11 6 24 26 10 37 + + + + + + + + + + 22.22.22.22.22. a a b ab b a b a b a b ab a a b a b a b ab b 5 3 2 4 5 4 3 2 2 3 4 5 4 3 2 2 3 4 5 26 8 6 8 15 45 8 8 27 15 53 14 + + + + + + + + 23.23.23.23.23. y y y y y y y y y y y y y y y y y y y 6 3 2 3 4 5 6 3 2 4 5 6 5 4 3 2 9 23 5 8 15 8 22 14 8 15 8 15 8 22 8 14 + + + + + + + + + + + + + + 24.24.24.24.24. x x x x x x x x x x x x x x x x x x x x x x x x 8 6 4 2 7 5 3 8 6 4 2 7 5 3 8 7 6 5 4 3 2 3 5 9 36 7 5 23 51 3 5 45 7 5 23 51 7 5 3 23 5 51 45 + + + + + + 25.25.25.25.25. x x y x y x y x y x y y x y xy x x y x y x y y x y xy x x y x y x y x y xy y 7 5 2 4 3 2 5 4 3 2 5 7 3 4 6 7 5 2 4 3 2 5 7 3 4 6 7 5 2 4 3 3 4 2 5 6 7 3 35 8 60 60 90 50 3 95 7 60 90 50 3 95 90 7 50 60 + + + + + + + + + + + 26.26.26.26.26. a a a a a a a a a a a a a x x x x x x x x x x x x x + + + + + + + + + + + + + + 3 1 1 2 3 1 2 3 2 1 8 5 5 6 3 5 6 3 6 5 27.27.27.27.27. + + + + + + + 8 16 15 7 5 8 15 16 10 8 15 8 10 16 4 2 3 2 3 1 4 2 1 1 2 4 a a a a a a a a a a a a a a a a n n n n n n n n n n n n n n n n 28.28.28.28.28. 15 5 6 41 9 18 31 15 14 6 59 31 15 14 31 6 59 3 2 1 2 1 1 4 3 2 1 1 4 4 3 2 1 1 x x x x x x x x x x x x x x x x x x x x a a a a a a a a a a a a a a a a a a a a + + + + + + + + + + + + + + + + + + + + + + + + 29.29.29.29.29. 9 21 26 14 5 12 8 14 33 26 14 8 14 33 26 8 14 1 2 3 5 1 2 4 1 2 3 5 4 1 2 3 4 5 a a a a a a a a a a a a a a a a a m m m m m m m m m m m m m m m m m m m m + + + + + + + + + + + + 30.30.30.30.30. + + + + + + + + + + + + + + + + + 15 50 14 6 8 23 6 15 23 56 14 5 8 3 1 1 2 4 2 1 1 4 3 2 1 1 2 m m m m m m m m m m m m m m m m x x x x x x x x x x x x x x x x
  23. 23. EJERCICIO 23 1.1.1.1.1. 1 1 2 + + a a 2.2.2.2.2. 0 8 8 + + a a 3.3.3.3.3. + 9 3 5 3 4 3 4 2 2 2 a a a a a a 4.4.4.4.4. + + + 16 5 16 5 5 2 2 2 xy x xy x x xy 5.5.5.5.5. 1 1 1 3 2 2 3 2 2 3 2 2 + + + + a a b ab a a b ab a a b ab 6.6.6.6.6. x x x y xy x x y xy 3 3 2 2 3 2 2 8 6 2 8 6 + + + + 7.7.7.7.7. a a b ab b a a b ab b 3 2 2 3 3 2 2 3 8 6 8 6 + + + + 8.8.8.8.8. + + + + y x y x y xy x y x y xy y 4 3 2 2 3 3 2 2 3 4 5 7 8 5 7 8 9.9.9.9.9. m m am a m a m a m am a m a m a 4 4 3 2 2 3 4 4 3 2 2 3 4 5 18 7 4 18 7 + + + + 10.10.10.10.10. 16 14 30 30 + + + b a c d b a c d a b c d 11.11.11.11.11. x xy y x xy y x xy y 2 2 2 2 2 2 1 1 1 12.12.12.12.12. a a b ab b a a b ab b 3 2 2 3 3 2 2 3 6 5 8 5 8 6 + + + + 13.13.13.13.13. x y x y xy x x y xy y 3 3 2 2 3 2 2 3 5 17 5 5 17 5 + + + + + + 14.14.14.14.14. x x y x y xy x x y x y xy 4 3 2 2 3 4 3 2 2 3 1 9 8 15 9 8 15 1 + + + + 15.15.15.15.15. a b a b a b a b ab a a b a b a b ab b a a b a b a b ab b 5 5 4 2 3 3 2 4 5 4 2 3 3 2 4 5 5 4 3 2 2 3 4 5 11 2 8 4 11 2 8 4 11 8 2 4 + + + + + + + + + 16.16.16.16.16. x x x x x x x x 4 2 3 4 3 2 50 5 25 5 25 50 + + + + + + 17.17.17.17.17. y y y y y y y y y y y y y y y y y y 6 5 4 3 2 6 5 4 3 2 6 5 4 3 2 41 9 17 18 41 9 17 18 9 17 18 41 + + + + + + 18.18.18.18.18. a a b a b a b a b ab b a a b a b a b a b ab b 6 4 2 2 4 5 3 3 5 6 6 5 4 2 3 3 2 4 5 6 9 15 17 14 15 9 17 14 + + + + + + + + + + 19.19.19.19.19. x x x x x x x x x x x x x 4 3 2 4 3 2 4 3 2 11 5 34 16 34 16 34 + + + + + + + + + 20.20.20.20.20. m m n mn n m m n mn n 3 2 2 3 3 2 2 3 1 7 3 7 3 1 + +
  24. 24. EJERCICIO 24 1.1.1.1.1. 1 2 1 4 1 3 2 5 2 1 4 1 3 2 5 3 4 1 3 2 5 2 2 2 2 2 2 2 a a ab b a ab b a ab b + + + = + 2.2.2.2.2. 15 4 5 2 3 5 9 4 5 2 3 135 5 9 4 5 2 3 140 9 4 5 2 3 15 5 9 + + + = + = + xy yz xy yz xy yz xy yz 3.3.3.3.3. 3 5 1 6 3 4 2 9 18 5 30 3 4 2 9 3 4 13 30 2 9 bc bc ab cd bc ab cd ab bc cd + + + + + + 4.4.4.4.4. 1 2 2 3 4 5 2 9 1 2 5 8 10 6 2 9 1 2 3 10 8 9 1 2 a b a b a b a b + + + = + 5.5.5.5.5. 5 9 3 8 1 10 5 7 3 11 5 9 30 8 80 5 7 3 11 5 9 38 80 5 7 3 11 5 9 5 7 19 40 3 11 2 2 2 2 2 2 2 2 2 x y y xy x y xy x y xy x xy y + + + = + + 6.6.6.6.6. 5 6 2 9 1 5 1 2 3 8 5 6 10 9 45 1 2 3 8 5 6 1 2 3 8 19 45 3 3 3 2 2 3 3 2 2 3 2 2 3 m n n m n mn m n m n mn m m n mn n + + + + + + + + 7.7.7.7.7. 3 7 1 3 3 5 5 14 1 2 1 8 6 5 14 2 3 6 1 14 1 6 3 5 1 8 2 2 2 2 2 2 a ab b a ab a ab a ab b + + + + 8.8.8.8.8. 3 8 5 6 1 10 3 5 3 10 2 15 24 40 50 18 60 1 20 10 39 40 68 60 21 10 39 40 17 15 21 10 2 2 2 2 2 2 2 2 2 2 x xy y x xy y x xy y x xy y x xy y + + + + + + = + = + 9.9.9.9.9. a a a a a a a a a a a 3 2 2 3 2 3 2 5 6 7 8 9 10 7 8 8 7 8 10 9 10 20 21 24 15 8 19 10 1 24 + + + + + + + = + 10 .10 .10 .10 .10 . m mn n mn n m n m mn n m n m m n mn n 3 2 3 2 3 2 3 2 3 2 3 2 2 3 7 12 4 7 5 9 5 21 1 8 21 20 36 4 7 7 5 21 1 8 5 21 1 36 11 7 1 8 + + + + + + + + + +
  25. 25. 12.12.12.12.12. 1 2 3 5 7 8 8 9 7 20 1 8 1 9 7 8 1 2 12 7 20 7 1 8 8 1 9 7 8 1 2 19 20 8 8 9 9 7 8 1 2 19 20 7 8 a b c d b c d a b c d a b c d a b c d + + + + + + + + + = + + = + + 6.6.6.6.6. 5 8 1 4 1 3 7 8 6 5 6 5 8 2 7 8 1 18 3 5 6 5 8 9 8 19 3 5 6 5 6 5 8 9 8 19 3 2 2 2 3 2 2 3 2 2 3 3 2 2 a b ab ab a a b ab a a b ab a a a b ab + + + + + = + + + 7.7.7.7.7. 2 11 5 14 1 3 6 7 8 2 9 2 11 20 49 56 3 2 9 6 2 11 29 56 5 9 6 3 2 2 3 4 2 2 3 4 3 2 2 3 4 3 2 2 3 m n m n mn m m n mn m m n m n mn m m n m n mn + + + + + + + = + + 8.8.8.8.8. + + + + + + + + + = + + = + + 7 8 1 14 2 3 1 3 7 1 2 3 7 1 8 2 9 1 2 7 8 1 6 14 2 3 8 3 24 63 2 9 1 2 7 8 5 14 2 3 11 24 65 9 1 2 7 8 5 14 2 3 11 24 7 4 3 2 2 3 4 5 3 2 4 5 4 3 2 2 3 4 5 4 3 2 2 3 4 5 4 3 2 2 3 4 2 9 x y x y x y xy x x y xy x x y x y x y xy x x y x y x y xy x x y x y x y xy 1.1.1.1.1. 3 8 5 6 5 6 9 20 24 5 6 11 24 5 6 2 2 2 2 a a a a a a a = 2.2.2.2.2. 8 6 5 1 2 3 5 16 1 2 30 3 5 5 15 2 33 5 5 a b a b a b a b + + + + = + 3.3.3.3.3. x x y x y x x y x x y 3 2 2 3 2 3 2 2 3 6 7 9 6 7 9 6 1 9 6 + + = 4.4.4.4.4. a b c a b c a b c a b c + + + + + = + 1 2 3 4 2 3 2 1 2 4 3 4 3 2 3 1 2 7 4 5 3 5.5.5.5.5. 2 3 5 6 1 2 2 3 3 5 6 6 1 2 2 1 3 1 6 3 2 m n p m n p m n p m n p + + + + + + = + 11.11.11.11.11. 3 5 3 4 5 7 2 3 5 8 1 3 5 6 3 5 5 3 4 5 8 15 7 21 4 5 6 2 5 3 4 5 8 8 21 1 6 4 3 3 4 4 2 2 3 4 4 3 2 2 3 4 4 3 2 2 3 4 x x y xy y x x y xy y x x y x y xy y x x y x y xy y + + + + + = + EJERCICIO 25
  26. 26. 9.9.9.9.9. 7 9 2 3 1 8 2 13 7 9 1 11 7 9 6 7 9 1 8 11 1 11 2 13 13 7 9 13 9 1 8 12 11 15 13 5 4 2 3 3 2 4 5 6 6 4 2 2 4 5 6 6 5 4 2 3 3 2 4 6 6 5 4 2 3 3 2 4 6 x y x y x y x y xy y x x y x y xy y x x y x y x y x y y x x y x y x y x y y + + + + + + + + + + + = + + + 10.10.10.10.10. 1 3 7 9 5 8 7 11 2 5 2 3 1 6 3 4 6 1 2 3 14 3 18 5 6 8 7 11 2 30 5 11 18 1 8 7 11 32 5 3 2 2 3 3 2 2 3 2 2 3 3 2 2 3 x x y xy y x x y xy x x y xy y x x y xy y + + + + + = 11.11.11.11.11. 3 10 3 7 5 9 2 13 7 20 5 14 1 3 3 5 2 13 6 7 20 6 5 14 5 3 9 3 5 2 13 13 20 11 14 2 9 3 5 4 2 2 4 6 6 4 2 2 4 6 6 4 2 2 4 6 6 4 2 2 4 6 m n m n n m m n m n n m m n m n n m m n m n n + + + + + + + + = + + + 12.12.12.12.12. 3 8 7 22 7 12 1 2 1 3 35 5 11 5 6 3 4 3 13 3 8 7 10 22 7 10 12 1 2 3 4 13 9 39 35 3 8 17 22 17 12 1 2 3 4 22 39 35 5 4 3 2 2 3 5 4 3 2 4 5 5 4 3 2 2 3 4 5 5 4 3 2 2 3 4 5 c c d c d c d d c d c d cd d c c d c d c d cd d c c d c d c d cd d + + + + + + + + + + + = + + + EJERCICIO 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 1.1.1.1.1. a ab ab b a ab b 2 2 2 2 2 2 3 4 1 4 1 2 2 1 8 4 11 = = = 2.2.2.2.2. a b b a b ab a b a b ab a a b ab b 3 3 3 2 2 3 3 2 2 3 2 2 3 3 2 2 3 2 5 6 3 5 6 5 6 3 1 5 1 2 6 1 2 3 2 1 5 2 6 4 3 8 1 10 24 24 11 + + + + + + + = + + = + + = + + =
  27. 27. 3.3.3.3.3. 1 2 1 2 5 3 1 2 2 1 2 5 3 1 2 1 2 5 3 1 2 1 1 2 2 5 3 3 1 2 1 5 1 2 10 2 7 2 3 1 2 a a b c a b c a b c + + = + + = + = + = = 4.4.4.4.4. 3 5 10 8 2 15 8 2 8 15 2 3 2 8 3 2 2 5 15 2 5 2 9 4 48 10 3 4 5 9 2 24 5 12 5 45 48 24 10 27 10 2 2 2 2 2 2 2 2 2 2 m n m n mn m n mn m mn n = = = = = = 5.5.5.5.5. x x y y x y xy y x x y x y xy y 4 2 2 4 3 3 4 4 3 2 2 3 4 4 3 2 2 3 4 18 15 16 6 9 16 18 6 6 4 16 4 5 18 4 5 6 4 5 6 5 256 80 64 18 16 25 24 125 6 625 256 5120 7 200 3 000 3 750 4 926 + + + + + = + + + = + + + = + + + =. . . . . 6.6.6.6.6. a am m am m a m a am m a m a a m am m 3 2 3 2 3 2 3 2 3 2 3 2 2 3 3 2 2 3 7 5 5 8 2 6 8 8 2 6 1 8 1 3 2 2 1 3 2 6 3 2 1 24 2 2 9 4 6 27 8 1 12 9 2 81 4 4 48 18 81 4 19 4 + + + + + = + = + = + = + = 7.7.7.7.7. 2 3 7 8 1 5 1 6 1 10 4 1 6 7 8 8 2 1 10 1 2 1 8 1 10 1 2 1 1 8 1 2 1 10 2 1 2 2 8 4 10 1 2 1 4 2 5 10 5 8 20 3 20 2 2 2 2 2 2 2 2 2 2 a ab b a ab b a ab b a ab b + + + = = = = = 8.8.8.8.8. 2 3 3 4 1 2 1 6 1 4 1 2 4 1 6 3 1 4 5 6 3 2 5 6 3 2 2 5 3 2 2 5 27 8 5 6 18 20 3 2 4 25 27 8 3 4 12 50 675 150 48 200 873 200 2 2 3 2 2 3 3 2 2 3 3 2 2 3 2 2 m n mn n m n mn n m m n mn m m m n mn + + + + + + + + = + + = + + = + + = + + = + + = 9.9.9.9.9. a a b b a b a b a a b a b a b b 5 2 4 5 4 2 3 3 5 4 2 3 3 2 4 5 5 4 2 3 3 2 4 5 3 5 5 3 1 1 2 5 1 2 3 1 2 2 1 4 5 8 3 16 32 1 4 40 48 32 21 + + + + = + + = + + = + + = 10.10.10.10.10. + + = + = + = ab mn mx ab ab mn mx 10 8 15 16 10 8 16 1 2 10 3 2 2 5 8 3 2 4 32 6 48 74 11.11.11.11.11. a a b ab b a a b ab b 3 2 2 3 3 2 2 3 3 2 2 3 11 9 11 9 1 11 1 2 9 1 2 2 1 22 36 8 7 + + = + = + = 12.12.12.12.12. 1 64 2 3 5 6 3 8 1 64 2 3 5 6 3 8 1 64 4 2 3 4 5 6 4 3 8 4 32 3 20 6 3 8 96 256 80 9 24 231 24 77 8 9 4 2 4 2 4 2 5 8 x x x x x x + + = + = + = + = = =
  28. 28. 13.13.13.13.13. x x y xy x xy y x x y xy y x x y xy y 3 2 2 3 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 16 2 5 3 4 3 5 1 25 4 3 4 3 16 2 3 5 1 25 1 4 3 16 1 5 1 25 1 4 4 3 16 4 5 1 5 4 5 1 25 5 1 4 64 15 16 16 4 5 25 125 25 16 15 20 5 56 + + + + + + + + = + + + = + + + = + + + = EJERCICIO 27 1.1.1.1.1. + + + + + ab b a a b a ab b a ab b ab b 2 2 2 2 2 2 2 2 2 5 4 4 4 2.2.2.2.2. a a + + 8 1 6 14 14 13 3.3.3.3.3. + + + + + + x xy x y y x x y xy y x y x x y xy y x x y xy y 3 2 2 3 3 2 2 3 2 3 2 2 3 3 2 2 3 4 5 5 4 7 5 4 12 4 4.4.4.4.4. + + + 3 4 5 3 4 5 4 4 5 4 3 2 3 4 3 2 3 3 3 4 3 m n mn n m m n mn n n n m n 5.5.5.5.5. 8 9 3 6 7 9 3 5 a b c a a b c a a a + + 6.6.6.6.6. a b c a b c a b c a a a b c + + + + 2 2 14.14.14.14.14. 2 5 5 6 9 2 5 5 5 54 6 3 5 49 6 1 3 5 1 49 6 1 1 3 5 49 6 30 18 245 30 257 30 8 1 3 2 1 3 2 1 3 1 3 4 4 1 4 3 17 30 a a a a a a a a a a a a a x x x x x x x x x x x x x + + + + + = + = + = + + = = 7.7.7.7.7. + + + + + m n p m n p m n p m n p m n p 2 2 2 0 8.8.8.8.8. 9 5 3 9 7 25 25 6 6 25 24 5 3 2 2 2 2 2 2 2 2 2 2 2 ax a x ax a ax a x x a a x x ax a + + + + 9.9.9.9.9. 5 6 4 1 2 8 6 2 5 2 2 2 5 2 2 5 2 3 2 3 3 3 2 3 2 3 2 a a a a a a a a a a a a a a + + + + + + 10.10.10.10.10. 5 9 4 1 11 7 6 11 2 9 6 4 11 2 9 6 4 12 2 9 6 5 3 2 4 4 3 4 3 2 4 3 2 4 3 2 x x x x x x x x x x x x x x x x x x + + + + + + + + 11.11.11.11.11. 35 7 11 7 35 8 6 7 5 7 5 8 5 2 2 3 3 3 2 2 3 2 3 2 3 2 3 a b ab a b a a b ab a ab a ab a ab b + + + + + + + 12.12.12.12.12. + + + + + + + + + + 11 14 25 8 19 6 9 4 11 19 8 16 4 7 4 11 19 8 16 4 11 26 8 20 4 4 2 3 2 4 3 2 5 3 4 3 2 5 4 3 2 n n n n n n n n n n n n n n n n n n n n n n
  29. 29. 13.13.13.13.13. + + + + + + + + + + + 6 5 6 7 5 11 6 7 11 11 5 6 6 8 7 11 11 5 6 6 6 11 3 5 7 6 3 3 4 3 2 2 4 4 3 2 2 3 4 4 2 2 4 4 3 2 2 3 4 4 3 2 2 3 4 a m am a a m a m m a a m a m am m a a m m a a m a m am m a a m a m am m 14.14.14.14.14. + + + + + + + + + + + + + 4 13 9 6 8 2 6 4 8 13 8 2 30 40 6 4 8 13 8 2 7 4 38 13 48 3 4 2 3 4 5 3 2 4 5 5 4 3 2 2 3 4 5 5 3 2 4 5 5 4 3 2 2 3 4 5 5 4 3 2 2 3 4 5 x y x y xy x x y xy y x x y x y x y xy y x x y xy y x x y x y x y xy y x x y x y x y xy y 15.15.15.15.15. a b a a b a b a b + + 2 2 2 16.16.16.16.16. 8 9 8 6 4 6 5 2 8 6 4 8 6 6 x x y y x y x y + + + + + + + 17.17.17.17.17. x y x xy y xy y y x xy y x xy y 2 2 2 2 2 2 2 2 2 2 6 7 34 7 40 9 16 7 34 7 43 16 + + + + 18.18.18.18.18. 4 8 5 5 7 6 4 5 2 2 2 2 2 2 2 2 2 2 2 2 2 a ab b a ab b a ab b a ab b a ab b a b + + + + + + + + 19.19.19.19.19. x y x y xy x x y xy y x x y xy y x y x x y xy y 3 3 2 2 3 2 2 3 3 2 2 3 3 3 3 2 2 3 14 5 14 5 14 5 3 19 4 14 5 20 + + + + 20.20.20.20.20. x x y y x x y y x y y x x y y x x y y 4 2 2 4 4 2 2 4 2 2 4 4 2 2 4 4 2 2 4 6 2 32 8 31 2 32 2 32 0 + + + + + + 21.21.21.21.21. + + + + + + + + + + + 6 7 8 6 6 7 8 6 6 7 8 6 3 8 19 6 4 15 8 25 5 4 2 3 2 5 4 3 5 4 3 6 4 3 6 5 4 3 n n n n n n n n n n n n n n n n n n n n n n 22.22.22.22.22. a a b ab a b a b ab b a a b a b ab b a a b a b ab b a b a b b a a b a b a b ab b 5 3 2 4 4 3 2 4 5 5 4 3 2 4 5 5 4 3 2 4 5 4 2 3 5 5 4 3 2 2 3 4 5 3 6 22 10 11 22 7 5 22 7 5 5 7 17 7 7 5 2 + + + + + + + + + + + 23.23.23.23.23. 4 5 2 3 5 6 4 7 8 4 5 3 5 6 4 3 5 6 1 3 2 3 2 3 4 3 2 4 3 2 m m m m m m m m m m m m m m m m + + + + + + + 24.24.24.24.24. 7 11 2 8 7 11 8 4 2 8 2 4 2 2 2 2 2 2 2 a ab b b a ab b b b + + + + 25.25.25.25.25. 3 4 5 7 8 11 5 6 2 11 2 7 2 5 6 2 11 6 7 11 a b c a b a b c a b c a b c a b c a b + + + + + + + + 26.26.26.26.26. 5 14 19 8 9 1 3 1 5 17 10 6 3 5 2 8 17 10 1 3 2 5 4 2 5 4 3 2 4 3 5 4 3 2 a a a a a a a a a a a a a a a a a a a + + + + + + + + + + +
  30. 30. 27.27.27.27.27. m m n n m n m n mn n m m n m n mn n m m n mn n m m n m n mn n m m n m n mn n 4 2 2 4 3 2 2 3 4 4 3 2 2 3 4 4 2 2 3 4 4 3 2 2 3 4 4 3 2 2 3 4 10 15 11 14 3 11 4 3 16 6 7 8 11 4 3 16 5 11 11 11 17 + + + + + + + + + + + + 28.28.28.28.28. a a b ab b a b a b ab b a b a b ab a a b a b a b ab b a a b ab a a b a b a b ab b a a b a b a b ab b 5 3 2 4 5 4 2 3 4 5 3 2 2 3 4 5 4 3 2 2 3 4 5 5 2 3 4 5 4 3 2 2 3 4 5 5 4 3 2 2 3 4 5 4 8 7 15 25 3 3 5 7 3 18 22 2 3 6 21 6 7 3 18 22 2 2 7 3 24 2 6 + + + + + + + + + + + + EJERCICIO 28 29.29.29.29.29. x y x y x y x y y x x y x y x y x x y x y x y xy y x x y x y x y x y x y xy y 5 5 4 3 2 2 3 5 5 4 3 2 2 3 5 4 3 2 2 3 4 5 5 4 3 2 2 3 4 3 2 4 5 3 21 18 3 21 18 32 26 18 2 3 21 18 29 47 2 + + + + + + + + + + 30.30.30.30.30. 3 6 7 4 8 7 12 4 8 7 5 13 1 1 2 1 2 2 1 1 1 2 2 1 1 2 a a a a a a a a a a a a a a a a a a a a x x x x x x x x x x x x x x x x x x x x + + + + + + + + + + 6.6.6.6.6. a x x a x ax x a ax a a x ax x a a x ax x a a x ax x a a x ax x a a x ax x a x ax 2 3 2 2 3 3 2 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 2 2 3 5 11 11 3 4 6 8 3 3 9 17 3 3 3 9 17 3 10 14 + + + + + + + + + + + 7.7.7.7.7. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 4 2 3 3 2 4 2 4 4 3 2 4 3 2 4 3 2 4 3 2 4 3 2 3 3 5 7 8 3 4 5 4 3 2 4 2 7 8 3 1 2 4 2 7 8 3 1 6 12 4 1 + + + + + + + + + + + + + + + + 1.1.1.1.1. x x x x x x x x x x 2 2 2 2 5 4 2 1 2 6 6 2 2 1 2 2 3 + + + + + 2.2.2.2.2. 3 5 7 4 6 2 3 8 3 7 7 3 4 6 2 7 7 3 3 a b c a b a b c a b c b c a b c a b c a b c a b c + + + + + + 3.3.3.3.3. x x x x x x x x x x x x x x x x x 3 3 2 3 2 2 2 3 2 2 3 2 1 9 4 6 8 5 7 3 1 3 8 5 6 8 3 8 5 6 2 8 3 + + + + + + + + + + 4.4.4.4.4. a a a a a a a a a a a a a a a a 2 4 3 2 3 4 3 2 4 4 3 2 1 2 1 2 + + + + + + + 5.5.5.5.5. ab bc ac ab bc ac ab bc ac bc ac ab bc ac ab ac ab bc ac ab ac ab bc + + + + + + + + + 5 3 4 6 9 9 7 8 9 3 5 4 9 6 9 9 4 9 3 6 9
  31. 31. 8.8.8.8.8. m n m n m n mn m m n m n mn n m m n n m n mn m m n mn m n m n mn n m m n mn m m n m n mn n 4 4 3 2 2 3 4 3 2 2 3 4 4 2 2 4 2 2 3 4 2 2 3 3 2 2 3 4 4 2 2 3 4 3 2 2 3 4 17 4 7 6 17 2 7 81 6 80 4 2 17 2 7 81 2 17 3 8 81 2 + + + + + + + + + + 9.9.9.9.9. + + + + + + + + + + + + + + + a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a 3 5 4 2 4 3 2 5 4 3 2 2 3 2 4 3 2 5 4 3 2 4 3 2 5 3 2 7 6 8 4 11 10 27 5 11 26 16 8 7 15 12 7 7 15 11 10 27 12 7 7 15 11 4 3 42 10.10.10.10.10. 3 3 20 11 8 11 9 14 19 8 9 3 11 8 14 3 11 8 14 20 11 8 17 14 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 x y x xy y x xy y xy y x xy y x xy y x xy y x xy y x + + + + + + + + 11.11.11.11.11. a a a a a a a + + + 1 3 1 4 0 3 8 2 1 2 2 12.12.12.12.12. a ab b a ab b a ab b a ab b a ab b ab b a ab b a ab b a ab b ab b 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 8 7 9 3 4 5 11 17 8 7 2 9 2 9 4 5 + + + + + + + + + + 13.13.13.13.13. m m m m m m m m m m m m m m m m m m m m m m m m m m m 4 3 2 5 5 4 2 2 4 2 4 3 2 4 3 2 5 4 2 5 4 3 1 8 6 5 16 16 7 19 7 1 16 7 3 7 13 5 7 13 5 16 7 19 17 13 24 + + + + + + + + + + + + 14.14.14.14.14. x y x y x y x y y x y x y xy x x y x y x y xy y x y xy xy x y x y xy x x y x y x y xy y x y x y xy x x y x y x y xy y 5 5 4 3 2 2 3 5 4 3 2 4 5 4 3 2 2 3 4 5 3 2 4 4 4 3 2 4 5 4 3 2 2 3 4 5 4 3 2 4 5 4 3 2 2 3 4 5 2 5 7 3 7 11 2 2 7 6 4 8 7 6 8 1 7 10 1 2 2 7 6 4 8 7 10 1 9 7 4 4 7 + + + + + + + + + + + + + + + + + 15.15.15.15.15. + + + + + + + + + + + + + + + + + + + + + + a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a 6 4 5 3 2 4 3 4 2 5 4 3 2 3 2 5 3 2 6 5 4 3 2 6 5 4 3 2 5 4 3 2 6 5 4 3 7 8 3 11 4 6 11 2 8 3 7 8 5 5 3 7 48 58 13 5 5 4 1 5 7 41 50 8 3 5 4 14 13 3 5 4 14 13 5 3 7 48 58 13 8 4 2 44 2 44 a 16.16.16.16.16. a a x a x x a x a x ax a x a x a x a x ax x a a a x a x a x ax x a a x a x a x x a a x a x a x x a a x a x a x ax x 5 3 2 3 2 5 4 2 3 4 4 3 2 2 3 3 2 4 5 5 5 4 3 2 2 3 4 5 5 4 3 2 2 3 5 5 4 3 2 2 3 5 5 4 3 2 2 3 4 5 7 9 18 4 8 20 21 19 9 17 11 9 7 80 36 20 2 21 26 71 9 11 4 28 9 11 4 28 20 2 21 26 71 11 + + + + + + + + + + + + + + + + + + + a x a x a x ax x4 3 2 2 3 4 5 10 26 5 99 + +
  32. 32. 1.1.1.1.1. a b a a b a b a b a b a b a b + + + + = + = 1 2 3 4 2 3 3 4 1 3 5 4 3 2 3 2 3 4 9 4 12 5 4 1 3 5 4 5 12 5 4 2.2.2.2.2. 3 8 6 1 2 3 5 5 6 3 5 5 6 3 5 3 8 6 5 6 3 5 3 8 6 3 5 6 3 8 6 4 3 3 8 6 3 2 3 2 3 2 3 2 3 3 a a a a a a a a a a a a a a a + + + + + + + = + 3.3.3.3.3. a b a b a b a b a b a b a b a b + + + + + + + + + + = + + = + + 3 3 5 7 3 6 2 5 2 3 6 1 5 1 6 5 2 5 9 2 3 6 3 1 5 14 1 6 6 3 5 7 3 6 2 5 5 2 6 4.4.4.4.4. 1 3 3 7 1 5 1 14 2 9 6 1 3 6 1 14 2 9 1 30 5 1 3 5 14 2 9 31 5 5 6 1 3 5 14 2 9 31 5 5 2 6 5 14 2 9 31 5 7 6 5 14 2 9 6 3 2 2 3 2 3 2 3 3 2 3 2 3 2 1 5 x x x x x x x x x x x x x x x x x x x x + + + + = + + + + + + = + + EJERCICIO 29 5.5.5.5.5. 7 12 7 12 3 7 2 5 6 3 7 2 5 6 3 4 1 5 1 3 7 12 3 7 2 5 6 7 9 12 3 7 2 5 1 5 18 1 3 4 3 3 7 2 5 1 5 17 3 4 4 3 2 3 2 4 4 3 2 4 3 2 4 3 2 a a a a a a a a a a a a a a a a a a a + + + + + + = + 6.6.6.6.6. + + + + + + + + = + + = + 1 2 2 3 1 4 5 9 2 5 1 9 2 5 3 1 2 5 9 13 20 3 1 2 6 1 9 5 8 20 3 1 2 13 20 2 15 5 1 2 5 9 13 20 3 1 2 13 20 17 5 x y z y y z x y z x y z x z x y z x z 7.7.7.7.7. + + + = + + + + 3 2 3 4 1 8 5 6 2 3 12 1 8 9 10 12 3 2 3 11 8 1 12 1 3 1 2 1 3 11 8 1 12 1 3 1 2 11 8 1 12 2 2 3 2 2 3 2 2 3 2 2 3 3 3 2 2 3 3 2 2 a b ab b a b ab b a b ab b a b ab b a b a b ab b a a b ab 8.8.8.8.8. 1 2 2 9 2 3 1 5 1 2 1 9 3 5 1 3 3 5 5 9 1 10 1 9 1 10 1 2 2 3 9 3 5 6 5 9 2 1 10 1 2 6 1 10 1 2 1 9 3 5 1 9 1 10 1 2 7 10 a b b c a b c b c b c b c a b c b c a c a b c b c a c + + + + = + = + =
  33. 33. 9.9.9.9.9. 1 3 1 8 1 5 1 4 2 3 1 4 1 3 19 40 2 3 1 8 3 5 3 4 1 10 1 3 29 40 1 8 1 3 19 40 3 4 1 10 1 3 5 24 40 3 4 2 1 10 1 3 10 29 40 2 3 2 1 8 8 9 12 10 8 80 1 3 19 40 3 4 1 10 1 3 19 40 2 3 1 8 1 12 2 80 1 12 1 40 3 2 2 3 2 2 3 2 3 2 3 2 3 2 3 2 3 2 a a a a a a a a a a a a a a a a a a a a a a a a a a a a a + + + + + + + + + + + = + = + = + = + 10.10.10.10.10. 3 5 5 6 2 9 2 9 1 9 2 3 3 5 7 3 13 6 1 2 3 2 1 3 1 4 17 45 22 9 3 2 1 2 3 5 7 3 1 9 1 4 3 5 10 18 12 2 3 9 1 4 10 17 45 1 22 9 4 9 6 1 2 39 2 18 2 1 4 3 5 28 12 1 9 1 4 27 45 21 9 13 6 1 2 37 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 x xy y x xy y x xy y xy y x xy y x xy y x xy y x xy y y x xy y x xy y y + + + + + + + + + + + + + + + = + = = + = + = 3 4 3 5 7 3 1 9 1 4 3 5 7 3 13 6 1 2 2 2 2 2 x xy y x xy y 11.11.11.11.11. 2 7 1 5 1 2 1 4 1 5 3 4 3 8 3 2 7 10 3 4 3 8 1 10 5 4 1 8 3 2 1 2 2 7 3 4 3 8 1 10 2 7 3 4 3 8 2 1 10 2 5 4 2 1 8 3 2 2 5 10 2 7 15 1 10 7 10 2 7 3 4 3 8 1 10 3 3 2 2 2 2 3 2 2 3 2 2 3 3 2 2 3 3 2 2 3 2 2 3 3 3 3 2 2 3 a b a b ab a b ab b a b ab b a b ab b a a b ab b a a b ab b a b ab b a b a a b ab b + + + + + + + + + + + = + = + = = 3 4 3 8 3 2 7 10 2 7 14 10 7 10 2 7 7 5 7 10 2 2 3 3 3 3 3 a b ab b a b a b 12.12.12.12.12. 1 3 1 4 2 7 3 5 2 5 5 3 5 14 2 5 1 14 7 20 1 4 2 3 5 14 1 4 11 60 1 4 4 1 14 12 7 20 20 24 15 60 1 4 3 5 2 3 1 4 11 60 1 4 2 5 5 14 5 20 11 60 1 4 0 3 2 2 3 4 4 3 2 2 4 4 4 4 3 2 2 4 4 3 2 2 3 4 3 2 2 3 4 3 2 2 3 4 4 3 2 2 3 m n mn n m m n m n n m n m m n m n n m m n m n mn m m n m n mn n m n m n mn n m m n m n mn n + + + + + + + + + + = + + 4 4 3 2 2 35 14 1 4 11 60 1 4 = + + m m n m n mn
  34. 34. 13.13.13.13.13. 1 2 1 3 3 4 1 6 2 5 1 4 5 1 2 1 3 3 8 1 2 13 12 7 30 1 4 1 3 3 8 1 2 4 9 12 5 12 30 1 2 4 1 3 3 8 1 2 13 12 7 30 1 4 1 3 40 3 8 1 2 13 12 7 30 1 4 1 3 3 8 1 2 13 12 7 30 1 4 1 3 37 8 x y y z z m m n x y z m n x y z m n x y z m n x y z m n x y z m n + + + + + + + + + + + + = + + + + = + + 14.14.14.14.14. 5 6 1 2 3 5 2 3 3 8 5 3 8 3 4 1 6 2 3 11 24 1 12 1 2 152 33 1 6 39 40 3 11 1 12 3 8 20 9 24 6 9 2 12 4 1 6 24 15 39 40 165 22 9 33 11 24 24 1 2 1216 99 264 11 24 1 12 3 6 0 40 152 33 11 24 1 12 1 2 152 33 4 3 2 4 3 2 4 3 2 3 4 3 4 3 2 4 2 4 3 2 4 3 2 a a a a a a a a a a a a a a a a a a a a a a a a a a a a + + + + + + + + + + + + + + = + = + = . + 13 24 1 2 1117 264 4 2 a a . EJERCICIO 30 1.1.1.1.1. x x x x x x Rta x x x 3 2 3 2 3 2 5 3 5 6 3 6 3 11 + + + + + . 4.4.4.4.4. x x x x x x Rta x x x 3 2 3 2 3 2 4 8 4 13 5 4 13 + + + +. 7.7.7.7.7. 5 8 11 4 8 11 5 8 11 3 2 3 3 3 3 2 3 2 3 a ab b a b a ab Rta a ab b + + + + . 2.2.2.2.2. + + + + + + + + 5 9 6 5 9 6 8 9 8 9 5 9 6 8 9 a b c a b c x x Rta a b c x. 5.5.5.5.5. m mn n m m n mn n m n Rta m m n mn n 4 3 4 4 2 2 3 4 2 2 4 2 2 3 4 3 6 4 3 6 8 4 8 4 3 6 8 + + + + +. 8.8.8.8.8. 1 2 1 3 4 1 2 1 3 4 x y x y + 3.3.3.3.3. a b a a b ab b a b ab b Rta a a b ab b 3 3 3 2 2 3 2 2 3 3 2 2 3 8 5 3 8 5 4 8 5 3 + + + . 6.6.6.6.6. 4 5 5 2 5 4 8 4 9 6 4 5 5 2 3 2 2 3 3 2 x x x x x x x Rta x x x + + + + . Rta x y. 1 2 1 3 4
  35. 35. 9.9.9.9.9. 5 7 8 5 7 8 1 1 5 7 8 1 2 2 2 2 2 2 x xy y x xy y Rta x xy y + + + + + +. 10.10.10.10.10. n m m n mn n m m n mn n m m n mn n m Rta m m n mn n 3 3 2 2 3 3 2 2 3 3 2 2 3 3 3 2 2 3 9 8 5 9 8 5 2 10 8 5 2 10 8 5 2 + + + + + + . 11.11.11.11.11. 0 5 8 5 8 0 3 3 + + a a a a Rta. EJERCICIO 31 1.1.1.1.1. ( )x x y x x y y = + = 6.6.6.6.6. ( ) ( )a a b a b a a b a b a a a + + + = + + = =2 11.11.11.11.11. ( ) ( )x y x y z x y z x y x y z x y z x y z + + + + = + + + + = + 2 2.2.2.2.2. ( )x x x x x x x 2 2 2 2 3 5 3 5 3 5 + + = + = + 7.7.7.7.7. a b a a b a b a a b a a a 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 + + = + + = = 12.12.12.12.12. ( ) ( ) ( )a b a a b a b a b a a b a b b + + + + = + + = 2 2 2 3.3.3.3.3. ( )a b a a b a a b + + = + + = + 2 3 2 3 3 3 8.8.8.8.8. { } { }2 1 3 2 1 3 1 3 4 a x a a x a x a a x + + = + + + = + = 13.13.13.13.13. ( ) ( ) [ ] + + + + = + + + + = + + x y xy x xy y xy x y xy x xy y xy x xy y 2 2 2 2 2 2 2 2 2 2 2 3 2 3 3 3 2 4.4.4.4.4. ( )4 2 4 2 6 m m n m m n m n = + + = + 9.9.9.9.9. ( ) [ ]x y x xy y x y x y x xy y x y x xy y 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + + + = + + = + 14.14.14.14.14. [ ] { } ( )8 2 3 3 8 2 3 3 8 4 2 2 2 2 2 2 2 2 2 2 2 2 x xy y x xy y x xy x xy y x xy y x xy x y + + + = + + + + = + 5.5.5.5.5. ( )2 3 4 3 2 3 4 3 2 x y x y x y x y x + + = + = 10.10.10.10.10.( ) ( ) + + + = + + = + 5 6 5 6 5 6 5 6 6 5 m m m m m 15.15.15.15.15. ( ) ( ) ( ) ( ) + + + + + = + + + = a b a b b a a b a b a b b a a b 3 3 0 EJERCICIO 32 1.1.1.1.1. ( )[ ] [ ] 2 2 2 2 a a a b a a a b a a a b a b + + = + = + = 2.2.2.2.2. ( )[ ] [ ] 3 2 3 2 3 2 4 x x y x y x x y x y x x y x y x + + = + = + + = 3.3.3.3.3. ( ) ( )[ ] [ ] 2 2 2 2 2 m m n m n m m n m n m m n m n m n + = = + + + = +
  36. 36. 4.4.4.4.4. ( ) ( ) ( )[ ] [ ] 4 3 2 3 4 3 2 3 4 3 2 3 6 3 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 x x xy y xy x y x x xy y xy x y x x xy y xy x y x xy y + + + + = + + + + = + + + = + 5.5.5.5.5. ( ) ( ){ } { } a a b a b c a a a b a b c a a a b a b c a a c + + + + = + + + + + = + + + + = + 2 2 2 6.6.6.6.6. ( )[ ] ( )[ ] [ ] [ ] 4 2 3 4 2 1 4 2 3 4 2 1 4 2 3 4 2 1 5 2 m m n n m m m n n m m m n n m n + + + = + + = + = + 7.7.7.7.7. { }( )[ ] ( )[ ] [ ] 2 5 2 2 5 2 2 5 2 2 5 2 2 x x y x y x x y x y x x y x y x x y x y x y + + + = + + = + + + = + + = + 8.8.8.8.8. ( )[ ]{ } [ ]{ } { } x xy y x xy y x xy y x xy y x xy y x xy y x xy y x xy y x xy y 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 7 3 2 7 3 2 7 3 2 7 3 2 2 4 3 + + + = + + = + = + + + + = + + 9.9.9.9.9. ( ) ( ){ }[ ] { }[ ] [ ] + + + + + = + + + + = + + + + = + + + = a b a b a b a b a a b a b a b a b a b a b a b a b a b a b a b a b a b 3 2 2 2 2 2 2 10.10.10.10.10. ( ) ( )[ ]{ } [ ]{ } { } + + + + = + + + = + + = + + + + + = + x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y x y 4 2 4 2 4 2 4 2 3 11.11.11.11.11. ( ) ( ) ( ) ( )[ ] [ ] + + + + + + = + + + = + + = a b a b a b b a b a b a b a b b a b a b a b a b b a b a b 2 3 2 3 2 3 3 7 12.12.12.12.12. ( ) ( )[ ]{ } ( ) [ ]{ } { } 7 3 5 3 2 3 7 3 5 3 2 3 7 3 5 3 2 3 7 3 5 3 2 3 7 2 5 2 2 2 2 2 2 2 2 2 2 2 2 2 m m n n m n m m n n m n m m n n m n m m n n m n m n + + + = + + + = + + = + + + + = + 13.13.13.13.13. ( ) ( ) ( )[ ]{ } [ ]{ } { } 2 4 4 2 4 4 6 4 6 4 a a b a b a b a a a b a b a b a a b a b a b a a b a b a b a b + + + = + + + = + + = + + = 14.14.14.14.14. ( ){ } ( )[ ]( ) { }[ ]( ) [ ]( ) ( ) 3 5 2 6 3 5 2 6 3 5 2 6 3 5 2 6 3 5 2 6 5 5 6 x y x y x x y x y x y x x y x y x y x x y x y x y x x y x y x y x x y x y + + + + = + + + = + + + = + + = + + + + = + 15.15.15.15.15. ( ) ( ) ( ){ }[ ] { }[ ] [ ] 6 2 2 2 6 2 2 2 6 2 2 2 6 2 2 2 6 7 c a c a c a a c c c a c a c a a c c c a c a c a a c c c a c a c a a c c a c + + + + + = + + = + = + + + + + + + = + 16.16.16.16.16. ( ) ( )( ){ } ( )[ ] ( ){ }[ ] { }[ ] [ ] + + + + = + + + = + + + = + + = + + + + = + + 3 2 2 2 5 6 3 2 2 2 5 6 3 2 2 2 5 6 3 2 2 2 5 6 3 2 2 2 5 6 6 2 1 m n m m m n n m n m m m n n m n m m m n n m n m m m n n m n m m m n n m n
  37. 37. 17.17.17.17.17. ( ) ( )( )[ ] ( ){ } ( )[ ]{ } [ ]{ } { } 2 5 3 2 4 2 5 3 2 4 2 5 3 2 4 2 5 3 2 4 2 5 3 2 4 5 6 a b a c a b c a b a b a c a b c a b a b a c a b c a b a b a c a b c a b a b a c a b c a b a b + + + + + + = + + + + + = + + + + + + + = + + + + = + + + = 18.18.18.18.18. ( )( )[ ] ( ) ( ) ( ){ } ( )[ ] { } [ ] + + + + + + = + + + + = + + = + + = 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 4 2 1 4 x x y x y x x y x x y x y x x y x x y x y x x y x x y x y 19.19.19.19.19. ( )[ ] ( )[ ] [ ] ( )[ ]{ } [ ] [ ] [ ]{ } { } + + + + = + = + + + = + = a a b c c a a b c c a a b c c b c c b 20.20.20.20.20. ( )[ ]{ } ( )[ ]{ } ( ) [ ]{ } [ ]{ } { } { } + + + = + + = + + = = a b b a a b a b b a a b a b b a a b a b b a a b a b3 3 21.21.21.21.21. ( )[ ]{ } ( )[ ]{ } ( ){ }[ ] [ ]{ } [ ]{ } { }[ ] { } { } [ ] + + + + + = + + + + = + + + + = + + + + + = + + a b c c a b a b a b c c a b a b a b c c a b a b a b c c a b a b a b c2 22.22.22.22.22. ( )( ){ } ( ) ( ){ }[ ] ( ){ } { }[ ] { }[ ] [ ] + + + + + + = + + + = + + + + = + + + = + + + + + = + 3 4 2 3 3 4 2 3 3 4 2 3 3 4 2 3 3 4 2 3 2 4 7 m m n m m n n m m n m m n n m m n m m n n m m n m m n n m m n m m n n m n 23.23.23.23.23. ( ) ( ) ( )[ ]{ }[ ] [ ]{ }[ ] { }[ ] [ ] + + + + = + + + = + + + + = + = + + = x x y x y z x y y x x y x y z x y y x x y x y z x y y x x y z x x y z y z 2 2 2 24.24.24.24.24. ( ) ( ) ( )[ ]{ }[ ] [ ]{ }[ ] { }[ ] [ ] + + + + = + + + + = + = = + + a a a b a b c a b a a a b a b c a b a a c a b a a c a b a b c3 EJERCICIO 33 1.1.1.1.1. ( ) a b c d a b c d + = + + 2.2.2.2.2. ( ) x xy y x xy y 2 2 2 2 3 6 3 6 + = + + 3.3.3.3.3. ( ) x x x x x x 3 2 3 2 4 3 1 4 3 1 + + = + + 4.4.4.4.4. ( ) a a b ab b a a b ab b 3 2 2 3 3 2 2 3 5 3 5 3 + = + + 5.5.5.5.5. ( ) x x x x x x x x 4 3 2 4 3 2 2 2 1 2 2 1 + + = + + 6.6.6.6.6. ( ) 2 2 a b c d a b c d + + = + 7.7.7.7.7. ( ) x x x x x x 3 2 3 2 3 4 3 4 + + = + 8.8.8.8.8. ( ) x x y xy y x x y xy y 3 2 2 3 3 2 2 3 5 3 5 3 + = + 9.9.9.9.9. ( ) a x xy y a x xy y 2 2 2 2 2 2 2 2 = + + 10.10.10.10.10. ( ) a b bc c a b bc c 2 2 2 2 2 2 2 2 + = + +
  38. 38. EJERCICIO 34 1.1.1.1.1. ( ) ( )[ ] x y x y x y x y + + = 2 2 2.2.2.2.2. ( ) ( ) ( ) ( )[ ] 4 2 3 2 4 2 3 2 m n m n m n m n m n m n + + + = + + 3.3.3.3.3. ( )[ ] ( )[ ]{ } x xy x xy y x xy x xy y 2 2 2 2 2 2 3 3 + + = + 4.4.4.4.4. [ ] ( ) [ ] ( )[ ] x x x x x x x x x x 3 2 3 2 3 4 2 3 2 3 3 4 2 3 2 3 + + + = + + + + 5.5.5.5.5. ( )[ ]{ } ( )[ ]{ }[ ] 2 3 2 2 3 2 a b a a b a a b a a b a + + + = + + + EJERCICIO 35 6.6.6.6.6. ( ) ( )[ ] + + = + 2 3 2 3 a a b a a b 7.7.7.7.7. ( ) ( ) ( ) ( )[ ] 2 3 2 3 2 2 2 2 2 2 2 2 x xy y xy x y x xy y xy x y + + + + = + + + 8.8.8.8.8. x x x x x x 3 2 3 2 3 4 2 3 4 2 + = + + { } 9.9.9.9.9. ( )[ ] ( ) ( )[ ] ( ){ } m m m m m m m m 4 2 4 2 3 2 3 2 3 3 2 3 2 3 + + + + = + + + 1.1.1.1.1. 2 3 6 = 2.2.2.2.2. =4 8 32 3.3.3.3.3. = 15 16 240 4.4.4.4.4. ab ab a b a b = = + +1 1 1 1 2 2 5.5.5.5.5. 2 3 6 62 2 1 3 x x x x = =+ 6.6.6.6.6. = =+ + 4 4 42 2 2 1 1 2 3 3 a b ab a b a b 7.7.7.7.7. = =+ + 5 5 53 2 3 1 1 2 4 3 x y xy x y x y 8.8.8.8.8. a b a x a b x a b x2 3 2 2 2 3 4 3 3 3 3 = =+ 9.9.9.9.9. = =+ 4 5 20 202 2 2 1 2 3 2 m mn p m n p m n p 10.10.10.10.10. 5 6 302 2 2 2 a y x a x y = 11.11.11.11.11. = =+ x y y z x y z x y z2 3 3 4 2 3 3 4 2 6 4 4 4 4 12.12.12.12.12. abc cd abc d abc d = =+1 1 2 13.13.13.13.13. = =+ 15 16 240 2404 3 2 3 2 4 3 3 2 7 3 x y a x a x y a x y 14.14.14.14.14. 3 4 122 3 2 2 3 2 a b x y a b x y = 15.15.15.15.15. 3 7 21 212 3 5 2 1 3 1 5 2 4 6 a bx b x a b x a b x = =+ + 16.16.16.16.16. = =+ 8 9 72 722 3 2 4 2 2 1 3 4 2 3 3 4 m n a mx a m n x a m n x 17.17.17.17.17. a b ab a bm n m n = + +1 1 18.18.18.18.18. = + + 5 6 302 3 2 3 a b a b x a b xm n m n 19.19.19.19.19. x y c x y c x y c x y cm n m n x m m n n x m n x = = + + + +1 2 2 1 20.20.20.20.20. = + + m n m n m nx a x a 6 62 2 1 EJERCICIO 36 1.1.1.1.1. a a a am m m m m = =+ + + +1 1 2 1 2.2.2.2.2. = =+ + + + x x x xa a a a a2 2 2 2 3.3.3.3.3. 4 4 41 1 1 1 2 1 a b ab a b a bn x x n x x n x = = + + + + + + 4.4.4.4.4. = = + + + + + + + + + + a b a b a b a bn n n n n n n n n n1 2 2 1 2 2 2 3 2 2 5.5.5.5.5. =+ + + + + + 3 4 124 1 2 3 2 6 2 4 a b a b a bn n n n n n 6.6.6.6.6. 3 4 12 122 3 1 2 2 1 3 2 3 5 x y x y x y x ym m m m m m = =+ + + + + + + + 7.7.7.7.7. 4 5 202 4 5 1 2 7 2 5 x b x b x ba a a a a a+ + + + + + = 8.8.8.8.8. a b c a b a b c a b cm n m n m m n n m n = = + +2 2 2 3 9.9.9.9.9. =+ + x y x y c x y cm a m a m a1 2 3 5 2 2 2 2 3 2 4 4 10.10.10.10.10. = 5 7 351 2 3 4 3 3 2 5 m n c m n m n ca b a b a b
  39. 39. EJERCICIO 37 1.1.1.1.1. 1 2 4 5 1 2 4 5 4 10 2 5 2 3 5 5 5 a a b a b a b a b = = = 2.2.2.2.2. = = = 3 7 7 14 3 7 7 14 21 98 3 14 2 2 3 2 5 2 5 2 5 m n a m a m n a m n a m n 3.3.3.3.3. 2 3 3 5 6 15 2 5 2 3 2 4 2 6 4 2 6 4 x y a x y a x y a x y = = 4.4.4.4.4. = = 1 8 4 5 4 40 1 10 3 4 3 2 3 5 5 3 5 5 m n a m n a m n a m n 5.5.5.5.5. = = 7 8 2 7 14 56 1 4 3 4 4 abc a a bc a bc 6.6.6.6.6. = = 3 5 5 6 3 6 1 2 3 4 2 5 2 3 9 2 3 9 x y a by a bx y a bx y 7.7.7.7.7. 1 3 3 5 3 15 1 5 1 1 a a a am m m = =+ + EJERCICIO 38 8.8.8.8.8. = =+ +3 4 2 5 3 2 1 5 3 10 3 1 3 1 3 a ab a b a bm m m 9.9.9.9.9. 5 6 3 10 1 2 1 2 1 4 2 1 2 1 2 a b ab c a b c a b c m n m n m n = = + + + + 10.10.10.10.10. = = + + + 2 9 3 5 2 3 1 5 2 15 1 1 2 1 2 1 2 1 2 1 a b a b a b a b x m x m x m x m 11.11.11.11.11. 3 8 4 5 12 40 3 10 2 3 2 3 2 a b a b a b a bm n m n m n m n = = 12.12.12.12.12. = = + 2 11 44 7 2 4 7 8 7 1 3 2 3 2 2 2 1 2 2 2 1 2 a b c a b a b c a b c x x x x x x x 1.1.1.1.1. a a a a a = = + + 3 3 32 1 1 2 4 2.2.2.2.2. 3 3 32 3 2 2 2 3 1 2 6 x x y a x a x y a x y = =+ + 3.3.3.3.3. = =+ + + m n m mn m n m n2 2 3 2 2 1 1 3 5 4 3 5 15 15 4.4.4.4.4. 4 5 20 202 3 2 2 2 3 1 2 2 6 2 2 a a x ay a x y a x y = =+ + 5.5.5.5.5. = = + + + + + a ab a b a b a bm x m x m x 2 3 6 62 1 2 1 3 1 6.6.6.6.6. 1 2 2 3 3 5 6 30 1 5 3 2 4 2 4 3 1 6 4 x a x a m a x m a x m = =+ + 7.7.7.7.7. 2 3 3 4 3 18 12 3 2 2 4 4 1 2 4 4 1 6 5 a a b a b a b a b m x m x m x = = + + + + + + + EJERCICIO 39 8.8.8.8.8. = = + + + + + 3 5 5 1 10 15 50 3 10 3 2 2 3 1 2 4 m a m a m a m a m x a x a x a 9.9.9.9.9. 2 3 4 24 242 3 1 2 3 1 7 a a a a a a = =+ + + 10.10.10.10.10. = = + + + + 3 4 5 60 60 2 3 2 3 1 2 2 1 1 6 4 b a b ab a x a b x a b x 11.11.11.11.11. a b a ab a x a b x a b x m x m x m x = = + + + + + + 2 2 2 1 2 1 5 1 2 3 6 6 12.12.12.12.12. = =+ + + + + 1 2 3 5 10 3 3 4 6 8 3 4 2 2 3 2 2 1 3 2 1 2 1 8 4 x y xy x x y x y x y 1.1.1.1.1. 3 2 6 2 3 2 4 3 x x x x x + 2.2.2.2.2. 8 3 2 16 6 2 2 3 5 3 2 x y y ax ax y ax y 3.3.3.3.3. x x x x x x 2 3 2 4 3 2 2 8 6 + + 4.4.4.4.4. a a a ab a b a b a b 3 2 4 3 2 4 6 3 3 12 18 + +
  40. 40. 5.5.5.5.5. a ab b ab a b a b ab 2 2 3 2 2 3 2 2 + + 6.6.6.6.6. x x x a x a x a x a x 5 3 2 2 2 7 2 5 2 3 6 8 3 3 18 24 7.7.7.7.7. m m n n m x m x m n x m n x 4 2 2 4 3 7 5 2 3 4 3 7 4 4 12 28 + + 8.8.8.8.8. x x y xy ax y ax y ax y ax y 3 2 2 3 6 5 2 4 3 4 6 4 6 + + 9.9.9.9.9. a a b ab a m a m a bm a b m 3 2 2 4 2 7 2 6 2 5 2 2 5 8 4 4 20 32 + + 10.10.10.10.10. a a a a a a a a a a m m m m m m m m m + + = + + + + + 1 2 1 1 1 2 1 1 1 2 2 2 2 2 2 2 11.11.11.11.11. x x x x x x x m m m m m m m + + + + 1 1 2 3 1 3 3 1 3 3 3 9 3 12.12.12.12.12. a b a b a b a b a b a b a b m n m n m n m n m n m n + + + + + + + + + 1 1 2 2 2 2 1 1 2 3 3 3 3 3 EJERCICIO 40 13.13.13.13.13. x x x x x x x x 3 2 2 5 4 3 2 3 5 6 4 4 12 20 24 + + + 14.14.14.14.14. a a x a x bx a bx a bx a bx bx 4 3 2 2 3 4 3 3 4 2 5 3 6 9 8 3 3 18 27 24 + + 15.15.15.15.15. a a a a a x a x a x a x a x n n n n n n n n n + + + + + + + + + 3 2 1 2 2 3 2 2 2 2 2 1 2 2 2 3 4 3 4 16.16.16.16.16. x x x x a x a x a x a x a x a x 4 3 2 2 3 2 7 2 6 2 5 2 4 2 3 6 8 7 5 3 3 18 24 21 15 + + + + 17.17.17.17.17. + + 3 5 7 4 5 15 25 35 20 3 2 2 3 2 2 2 4 2 2 3 3 2 2 4 2 5 x x y xy y a xy a x y a x y a x y a xy 18.18.18.18.18. x x x x x x x x x a a a a a a a a + + + + + + + + + + + 5 4 3 1 2 7 6 5 3 3 5 2 2 6 2 10 19.19.19.19.19. a a b a b a b b a y a y a b y a b y a b y a b y 8 6 2 4 4 2 6 8 3 2 11 2 9 2 2 7 4 2 5 6 2 3 8 2 3 3 5 5 15 5 15 5 + + + + 20.20.20.20.20. a b a b a b a b a b a b a b a b a b m n m n m n m n m m n m n m n m n + + + + + + + + + + + 3 4 4 12 4 4 1 2 2 4 3 6 3 2 3 2 1 5 2 2 7 2 3 9 1.1.1.1.1. 1 2 2 3 2 5 2 10 4 15 1 5 4 15 2 1 2 2 3 2 a b a a a b a a b = + 2.2.2.2.2. 2 3 3 4 2 3 4 9 6 12 4 9 1 2 3 3 1 3 1 1 4 3 2 a b a b a b a b a b a b + = + + + 3.3.3.3.3. 3 5 1 6 2 5 5 3 15 15 5 18 10 15 5 18 2 3 2 1 1 2 2 2 1 2 2 2 3 a b c ac a c abc ac a c abc ac + + = + + + 4.4.4.4.4. 2 5 1 3 2 9 3 6 5 3 3 6 9 6 5 2 3 2 2 2 2 2 2 1 2 2 4 3 2 2 a ab b a x a x a bx a b x a x a bx a b x + + = + + +
  41. 41. 5.5.5.5.5. 1 3 2 5 1 4 3 2 3 6 6 10 3 8 1 2 3 5 3 8 2 2 3 2 3 3 1 2 3 2 3 4 5 x xy y y x y xy y x y xy y = + + 6.6.6.6.6. 3 5 6 3 10 9 10 15 10 18 10 9 10 3 2 9 5 2 3 2 1 3 2 3 2 3 3 3 2 3 2 3 a b c a x a x a bx a cx a x a bx a cx + + = + + 7.7.7.7.7. 2 9 1 3 3 7 6 63 3 7 3 21 2 21 3 7 1 7 4 2 2 4 3 4 4 3 4 2 3 2 4 3 4 4 7 4 5 6 3 8 x x y y x y x y x y x y x y x y x y + + = + + + + + EJERCICIO 41 8.8.8.8.8. 1 2 1 3 1 4 1 5 5 8 5 16 5 24 5 32 5 40 5 16 5 24 5 32 1 8 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 a b x y a m a m a b m a mx a my a m a b m a mx a my + + + = + + + 9.9.9.9.9. 2 3 1 2 5 6 1 9 3 4 6 12 3 8 15 24 3 36 1 2 3 8 5 8 1 12 3 2 2 3 2 3 3 2 3 2 2 1 3 1 2 2 3 2 3 3 5 3 4 4 3 5 2 6 m m n mn n m n m n m n m n m n m n m n m n m n + + = + + + + + + + 10.10.10.10.10. 2 5 1 3 3 5 1 10 5 7 10 35 5 21 15 35 5 70 2 7 5 21 3 7 1 14 6 4 2 2 4 6 3 4 3 3 6 4 3 3 4 4 2 3 3 2 4 4 3 3 4 6 3 3 10 3 3 8 5 3 6 7 3 4 9 x x y x y y a x y a x y a x y a x y a x y a x y a x y a x y a x y + + + = + + + + + + + + 1.1.1.1.1. a a a a a a a + + + 3 1 3 3 2 3 2 2 3.3.3.3.3. x x x x x x x + + + 5 4 5 4 20 20 2 2 5.5.5.5.5. + + + + x x x x x x x 3 5 3 5 15 8 15 2 2 7.7.7.7.7. 3 2 2 6 4 3 2 6 2 2 2 2 2 x y x y x xy xy y x xy y + + 2.2.2.2.2. a a a a a a a + 3 1 3 3 2 3 2 2 4.4.4.4.4. m m m m m m m + + 6 5 6 5 30 11 30 2 2 6.6.6.6.6. + + + + + a a a a a a a 2 3 2 3 6 5 6 2 2 8.8.8.8.8. 5 4 3 2 15 12 10 8 15 22 8 2 2 2 2 x y x y x xy xy y x xy y + + + +