Ejercicio 140

CAPITULO XIV

Miscelania sobre Fracciones
Ejercicio 140
Simplificar:
  1. 12 x 2 +31x+20 18 x 2 +21x4 = 12 x 2 +16x+15x+20 18 x 2 3x+24x4 = 4x(3x+4 ) +5(3x+4 ) 3x(6x1 ) +4(6x1 ) = (4x+5 )(3x+4 ) (3x+4 )(6x1 ) = 4x+5 6x1
  2. ( 1 a + 2 a 2 + 1 a 3 ) ÷ (a+2 2a+1 a ) = 1 a (1+ 2 a + 1 a 2 ) ÷ [ a 2 +2a(2a+1 ) a ] = 1 a ( a 2 +2a+1 a 2 ) ÷ ( a 2 + 2a 2a 1 a ) = 1 a [ (a+1 ) 2 a 2 ] ÷ ( a 2 1 a ) = 1 a [ (a+1 ) 2 a 2 ] × [ a (a+1 )(a1 ) ] = a+1 a 2 (a1 )
  3. x 3 +3 x 2 +9x x 5 27 x 2 = x ( x 2 +3x+9 ) x 2 ( x 3 27 ) = x 2 +3x+9 x(x3 )( x 2 +3x+9 ) = 1 x(x3 )
  4. (x+y ) 2 y x (xy ) 2 xy = 1 y [ (x+y ) 2 x (xy ) 2 x ] = 1 y [ (x+y ) 2 (xy ) 2 ] = 1 y {[(x+y ) +(xy ) ] [(x+y ) (xy ) ] } = 1 y (x+ y +x y ) ( x +y x +y ) = 1 y (2x ) (2 y ) =4x
  5. a 4 2 b 3 + a 2 b(b2 ) a 4 a 2 b2 b 2 = a 4 2 b 3 + a 2 b 2 2 a 2 b a 4 + a 2 b2 a 2 b2 b 2 = a 4 + a 2 b 2 2 a 2 b2 b 3 a 2 ( a 2 +b ) 2b( a 2 +b ) = a 2 ( a 2 + b 2 ) 2b( a 2 + b 2 ) ( a 2 2b ) ( a 2 +b ) = ( a 2 2b )( a 2 + b 2 ) ( a 2 2b )( a 2 +b ) = a 2 + b 2 a 2 +b
  6. Multiplicar a+ 1+5a a 2 5 por a a+5 a+1
    (a+ 1+5a a 2 5 ) (a a+5 a+1 ) =[ a( a 2 5 ) +1+5a a 2 5 ] [ a(a+1 ) (a+5 ) a+1 ] =( a 3 5a +1+ 5a a 2 5 ) ( a 2 + a a 5 a+1 ) =( a 3 +1 a 2 5 ) ( a 2 5 a+1 ) = (a+1 )( a 2 a+1 ) a+1 = a 2 a+1
  7. Dividir x 2 +5x4 x 3 29 x5 entre x+34+ 170 x 2 x5
    ( x 2 +5x4 x 3 29 x5 ) ÷ (x+34+ 170 x 2 x5 ) =[ (x5 ) ( x 2 +5x4 ) ( x 3 29 ) x5 ] ÷ [ (x5 ) (x+34 ) +170 x 2 x5 ] =( x 3 + 5 x 2 4x 5 x 2 25x+20 x 3 +29 x5 ) ÷ ( x 2 +29x 170 + 170 x 2 x5 ) =( 4929x x5 ) ÷ ( 29x x5 ) =( 4929x x5 ) ( x5 29x ) = 4929x 29x
    Descomponer las expresiones siguientes en la suma o resta de tres fracciones simples irreducibles
  8. 4 x 2 5xy+ y 2 3x = 4 x 2 3 x 5 x y 3 x + y 2 3x = 4x 3 5y 3 + y 2 3x
  9. mnx mnx = m m nx n m n x x mn x = 1 nx 1 mx 1 mn
  10. Probar que x 3 x y 2 xy = x 2 +xy
    x 3 x y 2 xy = x 2 +xy x( x 2 y 2 ) xy =x(x+y ) x(x+y )(xy ) xy =x(x+y ) x(x+y ) =x(x+y )
  11. Probar que x 2 2x+1 9x3 x 2 x3 = x 3 1 x1
    x 2 2x+1 9x3 x 2 x3 = x 3 1 x1 (x3 ) ( x 2 2x+1 ) (9x3 x 2 ) x3 = (x1 )( x 2 +x+1 ) x1 (x3 ) ( x 2 2x+1 ) +3x(x3 ) x3 = x 2 +x+1 (x3 )[( x 2 2x+1 ) +3x ] x3 = x 2 +x+1 x 2 2x+1+3x = x 2 +x+1 x 2 +x+1 = x 2 +x+1
  12. Probar que a 4 5 a 2 +4 a 3 + a 2 4a4 =a3+ 2+4a 2a+1
    a 4 5 a 2 +4 a 3 + a 2 4a4 =a3+ 2+4a 2a+1 a 4 a 2 4 a 2 +4 a 2 (a+1 ) 4(a+1 ) = (2a+1 ) (a3 ) +2+4a 2a+1 a 2 ( a 2 1 ) 4( a 2 1 ) ( a 2 4 ) (a+1 ) = 2 a 2 +a6a3+2+4a 2a+1 ( a 2 4 )( a 2 1 ) ( a 2 4 )(a+1 ) = 2 a 2 a1 2a+1 (a+1 )(a1 ) a+1 = 2 a 2 2a+a1 2a+1 a1 = 2a(a1 ) +(a1 ) 2a+1 a1 = (2a+1 )(a1 ) 2a+1 a1 =a1
    Simplificar:
  13. 1 ab + 1 a+b + 2a a 2 ab+ b 2 = (a+b ) ( a 2 ab+ b 2 ) +(ab ) ( a 2 ab+ b 2 ) +2a(a+b ) (ab ) (ab ) (a+b ) ( a 2 ab+ b 2 ) = a 3 a 2 b+a b 2 + a 2 b a b 2 + b 3 + a 3 a 2 b+a b 2 a 2 b + a b 2 b 3 +2a( a 2 b 2 ) (ab ) ( a 3 + b 3 ) = 2 a 3 2 a 2 b+2a b 2 +2a( a 2 b 2 ) (ab ) ( a 3 + b 3 ) = 2a[ a 2 ab+ b 2 + a 2 b 2 ] (ab ) ( a 3 + b 3 ) = 2a[2 a 2 ab ] (ab ) ( a 3 + b 3 ) = 2 a 2 (2ab ) (ab ) ( a 3 + b 3 )
  14. ( a 2 1 a 2 a 4 1 a 4 ) × (1a+ 1+ a 3 a 2 ) =[ a 2 1 a 2 a 4 (1 a 2 ) (1+ a 2 ) ] × [ a 2 (1a ) +1+ a 3 a 2 ] = a 2 1 a 2 [1 a 2 1+ a 2 ] × [ a 2 a 3 +1+ a 3 a 2 ] = 1 1 a 2 ( 1+ a 2 a 2 1+ a 2 ) × ( a 2 +1 ) = 1 1 a 2
  15. ( x 2 9 x 2 x12 ÷ x3 x 2 +3x ) × a 2 x 2 16 a 2 2 x 2 +7x+3 × ( 2 a 2 x + 1 a 2 x 2 ) =[ (x3 ) (x+3 ) (x4 )(x+3 ) × x(x+3 ) x3 ] × a 2 ( x 2 16 ) 2 x 2 +x+6x+3 × 1 a 2 x (2+ 1 x ) = 1 x4 × x (x+3 ) × a 2 (x4 )(x+4 ) x(2x+1 ) +3(2x+1 ) × 1 a 2 x ( 2x+1 x ) = (x+3 ) × x+4 (x+3 ) (2x+1 ) × ( 2x+1 x ) = x+4 x
  16. Mathematical Equation
  17. 1681 x 2 72 x 2 5x12 = (49x ) (4+9x ) 72 x 2 32x+27x12 = (49x ) (4+9x ) 8x(9x+4 ) 3(9x+4 ) = (49x )(4+9x ) (8x3 )(9x+4 ) = 49x 8x3
  18. ( 1 x 2 x+2 + 3 x+3 ) ÷ ( x x+2 + x x+3 + 6 x 2 +5x+6 ) =[ (x+2 ) (x+3 ) 2x(x+3 ) +3x(x+2 ) x(x+2 ) (x+3 ) ] ÷ ( x x+2 + x x+3 + 6 (x+2 ) (x+3 ) ) =[ x 2 +5x+62 x 2 6x +3 x 2 + 6x x(x+2 ) (x+3 ) ] ÷ [ x(x+3 ) +x(x+2 ) +6 (x+2 ) (x+3 ) ] =[ 2 x 2 +5x+6 x(x+2 ) (x+3 ) ] ÷ [ x 2 +3x+ x 2 +2x+6 (x+2 ) (x+3 ) ] = 2 x 2 +5x+6 x (x+2 ) (x+3 ) × (x+2 ) (x+3 ) 2 x 2 +5x+6 = 1 x
  19. b a 1 b 2 a 2 + 1+ b ab 2 a3b ab = b a a 2 b 2 a 2 + a b b ab 2(ab ) (a3b ) ab = ab a 2 b 2 + a 2a2ba+3b = ab (a+b ) (ab ) + a a+b = a a+b [ b ab +1 ] = a a+b [ b +a b ab ] = a 2 a 2 b 2
  20. 1 3 ( x 2 36 x ÷ x x 2 4 ) × 1 x 36 x × 1 x 4 x = 1 3 [ (x+6 ) (x6 ) x × (x2 ) (x+2 ) x ] × 1 x 2 36 x × 1 x 2 4 x = 1 3 (x+6 ) (x6 ) x × (x2 ) (x+2 ) x × x x 2 36 × x x 2 4 = 1 3
  21. 3a (a2b ) 2 + 5 a5b + 1 a2b 3 a 2 14ab+10 b 2 a 2 4ab+4 b 2 = 3a(a5b ) +5 (a2b ) 2 +(a5b ) (a2b ) (a5b ) (a2b ) 2 3 a 2 14ab+10 b 2 (a2b ) 2 = 3 a 2 15ab+5( a 2 4ab+4 b 2 ) + a 2 7ab+10 b 2 (a5b ) (3 a 2 14ab+10 b 2 ) = 3 a 2 22ab+5 a 2 20ab+20 b 2 + a 2 +10 b 2 (a5b ) (3 a 2 14ab+10 b 2 ) = 9 a 2 42ab+30 b 2 (a5b ) (3 a 2 14ab+10 b 2 ) = 3 (3 a 2 14ab+10 b 2 ) (a5b )(3 a 2 14ab+10 b 2 ) = 3 a5b
  22. x+1 x1 x1 x+1 x1 x+1 + x+1 x1 × x 2 +1 2 a 2 2b ÷ 2x a 2 b = (x+1 ) 2 (x1 ) 2 (x1 ) (x+1 ) (x1 ) 2 + (x+1 ) 2 (x1 ) (x+1 ) × x 2 +1 2 ( a 2 b ) × a 2 b 2x = (x+1 ) 2 (x1 ) 2 (x1 ) 2 + (x+1 ) 2 × x 2 +1 2 × 1 2x = [(x+1 ) +(x1 ) ] [(x+1 ) (x1 ) ] x 2 2x +1+ x 2 + 2x +1 × x 2 +1 4x = (x+ 1 +x 1 ) ( x +1 x +1 ) 2 x 2 +2 × x 2 +1 4x = 2(2x ) 2 ( x 2 +1 ) × x 2 +1 4x = 1 2
  23. 1 3x9 1 6x+12 1 2 (x3 ) 2 + 1 x6+ 9 x = 1 3(x3 ) 1 6(x+2 ) 1 2 (x3 ) 2 + 1 x(x6 ) +9 x = 1 3(x3 ) 1 6(x+2 ) 1 2 (x3 ) 2 + 1 x 2 6x+9 x = 1 3(x3 ) 1 6(x+2 ) 1 2 (x3 ) 2 + x x 2 6x+9 = 1 3(x3 ) 1 6(x+2 ) 1 2 (x3 ) 2 + x (x3 ) 2 = 2(x+2 ) (x3 ) (x3 ) 2 3(x+2 ) +6x(x+2 ) 6(x+2 ) (x3 ) 2 = 2( x 2 x6 ) ( x 2 6x+9 ) 3x6+6 x 2 +12x 6(x+2 ) (x3 ) 2 = 2 x 2 2x12 x 2 +6x96+6 x 2 +9x 6(x+2 ) (x3 ) 2 = 7 x 2 +13x27 6(x+2 ) (x3 ) 2
  24. ab+ a 2 + b 2 a+b a+b a 2 2 b 2 ab × b+ b 2 a ab × 1 1+ 2ab b = (a+b ) (ab ) + a 2 + b 2 a+b (ab ) (a+b ) ( a 2 2 b 2 ) ab × ab+ b 2 a ab × 1 b +2a b b = a 2 b 2 + a 2 + b 2 a+b a 2 b 2 a 2 +2 b 2 ab × b(a+b ) a ab × b 2a = 2 a 2 a+b b 2 ab × b(a+b ) a(ab ) × b 2a = 2 a 2 (ab ) b 2 (a+b ) × b (a+b ) (ab ) × b 2 a =1

Ejercicio 139

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 139
Hallar el verdadero valor de:
  1. x2 x+3 parax =2 x2 x+3 = 22 2+3 = 0 5 =0
  2. x2 x3 parax =3 x2 x3 = 32 33 = 1 0 =
  3. x 2 a 2 x 2 + a 2 parax =a x 2 a 2 x 2 + a 2 = a 2 a 2 a 2 + a 2 = 0 2 a 2
  4. x 2 + y 2 x 2 y 2 parax =y x 2 + y 2 x 2 y 2 = y 2 + y 2 y 2 y 2 = 2 y 2 0 =
  5. x1 3 x2 parax =2 x1 3 x2 = (x1 ) (x2 ) 3 = x 2 3x+2 3 = 2 2 3( 2 ) +2 3 = 4 6 + 2 3 = 0 3 =0
  6. x 2 9 x 2 +x12 parax =3 x 2 9 x 2 +x12 = (x+3 )(x3 ) (x+4 )(x3 ) = x+3 x+4 = 3+3 3+4 = 6 7
  7. a 2 a6 a 2 +2a15 paraa =3 a 2 a6 a 2 +2a15 = (a3 )(a+2 ) (a+5 )(a3 ) = a+2 a+5 = 3+2 3+5 = 5 8
  8. x 2 7x+10 x 3 2 x 2 x+2 parax =2 x 2 7x+10 x 3 2 x 2 x+2 = (x5 ) (x2 ) x 2 (x2 ) (x2 ) = (x5 )(x2 ) ( x 2 1 )(x2 ) = x5 x 2 1 = 25 2 2 1 = 3 3 =1
  9. x 2 2x+1 x 3 2 x 2 x+2 parax =1 x 2 2x+1 x 3 2 x 2 x+2 = (x1 ) 2 x 2 (x2 ) (x2 ) = (x1 ) 2 ( x 2 1 ) (x2 ) = (x1 ) 2 (x+1 )(x1 )(x2 ) = x1 (x+1 ) (x2 ) = 1 1 (1+1 ) (12 ) = 0 2(1 ) =0
  10. a 3 8 a 2 +11a26 paraa =2 a 3 8 a 2 +11a26 = (a2 )( a 2 +2a+4 ) (a+13 )(a2 ) = a 2 +2a+4 a+13 = 2 2 +2( 2 ) +4 2+13 = 4+4+4 2+13 = = 4 5
  11. x 2 7x+6 x 2 2x+1 parax =1 x 2 7x+6 x 2 2x+1 = (x6 )(x1 ) (x1 ) 2 = x6 x1 = 16 11 = 5 0 =
  12. Mathematical Equation
  13. x 2 16 x 3 4 x 2 x+4 parax =4 x 2 16 x 3 4 x 2 x+4 = (x4 ) (x+4 ) x 2 (x4 ) (x4 ) = (x4 )(x+4 ) ( x 2 1 )(x4 ) = x+4 (x1 ) (x+1 ) = 4+4 (41 ) (4+1 ) = 8 3( 5 ) = 8 15
  14. 4 x 2 4x+1 4 x 2 +8x5 parax = 1 2 4 x 2 4x+1 4 x 2 +8x5 = (2x1 ) 2 4 x 2 2x+10x5 = (2x1 ) 2 2x(2x1 ) +5(2x1 ) = (2x1 ) 2 (2x+5 )(2x1 ) = 2x1 2x+5 = 2 ( 1 2 ) 1 2 ( 1 2 ) +5 = 11 1+5 =0
  15. Mathematical Equation
  16. Mathematical Equation
  17. x 3 a 3 xa parax =a x 3 a 3 xa = (xa )( x 2 +ax+ a 2 ) (xa ) = x 2 +ax+ a 2 = a 2 +a( a ) + a 2 =3 a 2
  18. a 2 2ab+ b 2 a 2 ab parab =a a 2 2ab+ b 2 a 2 ab = (ab ) 2 a (ab ) = ab a = a a a =0
  19. x 2 y 2 xy y 2 y =x x 2 y 2 xy y 2 = (xy )(x+y ) y (xy ) = x+y y = x+x x = 2 x x =2
  20. x 3 a 3 a 2 x a 3 parax =a x 3 a 3 a 2 x a 3 = (xa )( x 2 +ax+ a 2 ) a 2 (xa ) = x 2 +ax+ a 2 a 2 = a 2 +a( a ) + a 2 a 2 = 3 a 2 a 2 =3
  21. Mathematical Equation
  22. Mathematical Equation
  23. Mathematical Equation
  24. Mathematical Equation
  25. Mathematical Equation
  26. 8 x 2 +6x9 12 x 2 13x+3 parax = 3 4 8 x 2 +6x9 12 x 2 13x+3 = 8 x 2 +12x6x9 12 x 2 4x9x+3 = 4x(2x+3 ) 3(2x+3 ) 4x(3x1 ) 3(3x1 ) = (4x3 )(2x+3 ) (4x3 )(3x1 ) = 2x+3 3x1 = 2 ( 3 ) +3 3( 3 4 ) 1 = 3 2 +3 9 4 1 = 3+6 2 94 = 9 5 2 = 18 5
  27. x 3 +6 x 2 +12x+8 x 4 8 x 2 +16 parax =2 x 3 +6 x 2 +12x+8 x 4 8 x 2 +16 = ( x 3 +8 ) +(6 x 2 +12x ) ( x 2 4 ) 2 = (x+2 ) ( x 2 2x+4 ) +6x(x+2 ) [(x2 ) (x+2 ) ] 2 = (x+2 )[( x 2 2x+4 ) +6x ] (x2 ) 2 (x+2 ) 2 = x 2 +4x+4 (x2 ) 2 (x+2 ) = (x+2 ) 2 (x2 ) 2 (x+2 ) = x+2 (x2 ) 2 = 2 + 2 (22 ) 2 =0
  28. 9 x 3 +3 x 2 +3x+1 27 x 3 +1 parax = 1 3 9 x 3 +3 x 2 +3x+1 27 x 3 +1 = 3 x 2 (3x+1 ) +(3x+1 ) (3x+1 ) (9 x 2 3x+1 ) = (3 x 2 +1 )(3x+1 ) (3x+1 )(9 x 2 3x+1 ) = 3 x 2 +1 9 x 2 3x+1 = 3 ( 1 3 ) 2 +1 9 ( 1 3 ) 2 3 ( 1 3 ) +1 = 3 ( 1 3 2 ) +1 9 ( 1 3 2 ) +1+1 = 1 3 +1 3 = 1+3 3 3 = 4 9
  29. 1 x1 3 x 3 1 parax =1 1 x1 3 x 3 1 = x 2 +x+13 (x1 ) ( x 2 +x+1 ) = x 2 +x2 (x1 ) ( x 2 +x+1 ) = (x+2 )(x1 ) (x1 )( x 2 +x+1 ) = x+2 x 2 +x+1 = 1+2 1 2 +1+1 = 3 3 =1
  30. ( x 2 +3x10 ) (1+ 1 x2 ) parax =2 ( x 2 +3x10 ) (1+ 1 x2 ) =(x+5 )(x2 )( x2+1 x2 ) =(x+5 ) (x1 ) =(2+5 ) (21 ) =7( 1 ) =7

Ejercicio 138

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 138
Simplificar:
  1. 1+ x+1 x1 1 x1 1 x+1 = x 1 +x+ 1 x1 x +1 x +1 (x+1 )(x1 ) = 2 x(x+1 ) 2 =x(x+1 )
  2. 1 x1 + 2 x+1 x2 x + 2x+6 x+1 = x+1+2(x1 ) (x+1 )(x1 ) (x2 ) (x+1 ) +x(2x+6 ) x (x+1 ) = x+1+2x2 x1 x 2 x2+2 x 2 +6x x = 3x1 x1 3 x 2 +5x2 x = 3x1 x1 3 x 2 +6xx2 x = 3x1 x1 3x(x+2 ) (x+2 ) x = 3x1 x1 (3x1 )(x+2 ) x = x (x1 ) (x+2 )
  3. a ab b a+b a+b ab + a b = a(a+b ) b(ab ) (ab )(a+b ) b(a+b ) +a(ab ) b (ab ) = a 2 + ab ab + b 2 a+b ab + b 2 + a 2 ab b = b ( a 2 + b 2 ) ( a 2 + b 2 )(a+b ) = b a+b
  4. x+3 x+4 x+1 x+2 x1 x+2 x3 x+4 = (x+3 ) (x+2 ) (x+4 ) (x+1 ) (x+4 ) (x+2 ) (x1 ) (x+4 ) (x3 ) (x+2 ) (x+4 ) (x+2 ) = x 2 +5x+6( x 2 +5x+4 ) x 2 +3x4( x 2 x6 ) = x 2 + 5x +6 x 2 5x 4 x 2 +3x4 x 2 +x+6 = 2 4x+2 = 2 2 (2x+1 ) = 1 2x+1
  5. m 2 n m 2 n 2 m+n mn n + n m = m 2 (m+n ) n( m 2 n 2 ) n (m+n ) m(mn ) + n 2 n m = m 3 + m 2 n m 2 n + n 3 m+n m 2 mn+ n 2 m = m ( m 3 + n 3 ) (m+n ) ( m 2 mn+ n 2 ) =m
  6. a 2 b 3 + 1 a a b ba ab = a 3 + b 3 a b a(ab ) b(ba ) b (ab ) = a 3 + b 3 a b 2 a 2 ab b 2 + ab ab = ( a 3 + b 3 ) (ab ) a b 2 ( a 2 b 2 ) = (a+b )( a 2 ab+ b 2 )(ab ) a b 2 ( a 2 b 2 ) = a 2 ab+ b 2 a b 2
  7. 1+ 2x 1+ x 2 2x+ 2 x 5 +2 1 x 4 = 1+ x 2 +2x 1+ x 2 2x(1 x 4 ) +2 x 5 +2 1 x 4 = (x+1 ) 2 1+ x 2 2x 2 x 5 + 2 x 5 +2 (1+ x 2 )(1 x 2 ) = (x+1 ) 2 (1 x 2 ) 2x+2 = (x+1 ) 2 (1 x 2 ) 2 (x+1 ) = (x+1 ) (1 x 2 ) 2
  8. x+y xy xy x+y x+y x x+2y x+y = (x+y ) 2 (xy ) 2 (xy )(x+y ) (x+y ) 2 x(x+2y ) x (x+y ) = [(x+y ) (xy ) ] [(x+y ) +(xy ) ] xy x 2 + 2xy + y 2 x 2 2xy x = ( x +y x +y ) (x+ y +x y ) xy y 2 x = (2 y ) (2x ) xy y 2 x = 4 x 2 y(xy )
  9. a+x ax b+x bx 2 ax 2 bx = (a+x ) (bx ) (ax ) (b+x ) (ax ) (bx ) 2(bx ) 2(ax ) (ax ) (bx ) = abax+bx x 2 (ab+axbx x 2 ) 2(b x a+ x ) = ab ax+bx x 2 ab ax+bx+ x 2 2(ba ) = 2bx2ax 2(ba ) = 2 x (ba ) 2 (ba ) =x
  10. a a+x a 2a+2x a ax + a a+x = a a+x [1 1 2 ] a[ 1 ax + 1 a+x ] = a a+x [ 21 2 ] a[ a+ x +a x (a+x ) (ax ) ] = a 2 (a+x ) 2 a 2 (a+x )(ax ) = ax 4a
  11. a+2b ab + b a a+b a + 3b ab = a(a+2b ) +b(ab ) a(ab ) (a+b ) (ab ) +3ab a(ab ) = a 2 +2ab+ab b 2 a 2 b 2 +3ab = a 2 +3ab b 2 a 2 +3ab b 2 =1
  12. 1 7 x + 12 x 2 x 16 x = x 2 7x+12 x 2 x 2 16 x = (x3 )(x4 ) x (x4 )(x+4 ) = x3 x(x+4 )
  13. a 2 b b 2 a 1 b + 1 a + b a 2 = a 3 b 3 a b a 2 +ab+ b 2 a 2 b = a( a 3 b 3 ) a 2 +ab+ b 2 = a(ab )( a 2 +ab+ b 2 ) a 2 +ab+ b 2 =a(ab )
  14. x2y 4 y 2 x+y x3y 5 y 2 x+y = (x2y ) (x+y ) 4 y 2 x+y (x3y ) (x+y ) 5 y 2 x+y = x 2 xy2 y 2 4 y 2 x 2 2xy3 y 2 5 y 2 = x 2 xy6 y 2 x 2 2xy8 y 2 = x 2 +2xy3xy6 y 2 x 2 4xy+2xy8 y 2 = x(x+2y ) 3y(x+2y ) x(x4y ) +2y(x4y ) = (x3y )(x+2y ) (x+2y )(x4y ) = x3y x4y
  15. 2 1a + 2 1+a 2 1+a 2 1a = 2 [ 1 1a + 1 1+a ] 2 [ 1 1+a 1 1a ] = 1+ a +1 a (1a ) (1+a ) 1a(1+a ) (1a ) (1+a ) = 2 1 a 1 a = 2 2 a = 1 a
  16. 1 x+y+z 1 xy+z 1 xy+z 1 x+y+z = xy+z(x+y+z ) (x+y+z ) (xy+z ) x+y+z(xy+z ) (x+y+z ) (xy+z ) = x y+ z x y z x +y+ z x +y z = 2y 2y =1
  17. 1+ 2b+c abc 1 c2b ab+c = ab c +2b+ c abc ab+c(c2b ) ab+c = a+b abc ab+ c c +2b ab+c = a+b abc a+b ab+c = ab+c abc
  18. a 1a + 1a a 1a a a 1a = a 2 + (1a ) 2 a(1a ) (1a ) 2 a 2 a(1a ) = a 2 +12a+ a 2 12a+ a 2 a 2 = 2 a 2 2a+1 12a
  19. x+1 6x+12 x+2 x5 x4+ 11x22 x2 x+7 = (x+1 ) (x+2 ) (6x+12 ) x+2 x5 (x4 ) (x2 ) +11x22 x2 x+7 = x 2 3x10 x+2 x5 x 2 +5x14 x2 x+7 = (x5 ) (x+2 ) x+2 x5 (x+7 ) (x2 ) x2 x+7 =1
  20. 1 1+ 1 x = 1 x+1 x = x x+1
  21. 1 1+ 1 1 1 x = 1 1+ 1 x1 x = 1 1+ x x1 = 1 x1+x x1 = 1 2x1 x1 = x1 2x1
  22. 1 1 2+ 1 x 3 1 =1 1 2+ 1 x3 3 =1 1 2+ 3 x3 =1 1 2x6+3 x3 =1 x3 2x3 = 2x3(x3 ) 2x3 = 2x 3 x+ 3 2x3 = x 2x3
  23. 2 1+ 2 1+ 2 x = 2 1+ 2 x+2 x = 2 1+ 2x x+2 = 2 x+2+2x x+2 = 2 3x+2 x+2 = 2(x+2 ) 3x+2
  24. 1 x x x x 2 x+1 = 1 x x x(x+1 ) x 2 x+1 = 1 x x x 2 +x x 2 x+1 = 1 x x (x+1 ) x = 1 x x 1 =1
  25. 1 a+2 a+1 a 1 a = 1 a+2 a+1 a 2 1 a = 1 a+2 a(a+1 ) a 2 1 = 1 a+2 a (a+1 ) (a+1 )(a1 ) = 1 a+2 a a1 = 1 (a+2 ) (a1 ) a a1 = 1 a 2 + a 2 a a1 = a1 a 2 2
  26. x1 x+2 x 2 +2 x x2 x+1 = x1 x+2 x 2 +2 x(x+1 ) (x2 ) x+1 = x1 x+2 x 2 +2 x 2 + x x +2 x+1 = x1 x+2 x 2 +2 x 2 +2 x+1 = x1 x+2(x+1 ) = x1 x +2 x 1 =x1

Ejercicio 137

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 137
Simplificar:
  1. a a b b 1 b = aba b b 2 1 b = a(b1 ) b 2 1 = a (b1 ) (b1 )(b+1 ) = a b+1
  2. x 2 1 x 1 1 x = x 3 1 x x1 x = (x1 )( x 2 +x+1 ) x1 = x 2 +x+1
  3. a b b a 1+ b a = a 2 b 2 a b a+b a = a 2 b 2 b(a+b ) = (ab )(a+b ) b (a+b ) = ab b
  4. 1 m + 1 n 1 m 1 n = n+m mn nm mn = n+m nm
  5. x+ x 2 x x 4 = 2x+x 2 4xx = 2 (3x ) 3x =2
  6. x y y x 1+ y x = x 2 y 2 x y x+y x = x 2 y 2 y(x+y ) = (xy )(x+y ) y (x+y ) = xy y
  7. x+4+ 3 x x4 5 x = x(x+4 ) +3 x x(x4 ) 5 x = x 2 +4x+3 x 2 4x5 = (x+1 )(x+3 ) (x+5 )(x+1 ) = x+3 x+5
  8. a4+ 4 a 1 2 a = a(a4 ) +4 a a2 a = a 2 4a+4 a2 = (a2 ) 2 a2 =a2
  9. 2 a 2 b 2 a b 4 a 2 + b 2 4ab +1 = 2 a 2 b 2 ab a 4 a 2 + b 2 +4ab 4 a b = 2 a 2 ab b 2 4 a 2 +4ab+ b 2 = 2 a 2 2ab+ab b 2 (2a+b ) 2 = 2a(ab ) +b(ab ) (2a+b ) 2 = (2a+b )(ab ) (2a+b ) 2 = ab 2a+b
  10. 2+ 3a 5b a+ 10b 3 = 10b+3a 5b 3a+10b 3 = 3 5b
  11. ax+ x 2 a+x a 2 a 2 a+x = (ax ) (a+x ) + x 2 a+x a 2 (a+x ) a 2 a+x = a 2 x 2 + x 2 a 2 [(a+x ) 1 ] = a 2 a 2 (a+x1 ) = 1 a+x1
  12. a+5 14 a 1+ 8 a + 7 a 2 = a(a+5 ) 14 a a 2 +8a+7 a 2 = a( a 2 +5a14 ) (a+7 ) (a+1 ) = a (a+7 )(a2 ) (a+7 )(a+1 ) = a(a2 ) a+1
  13. 1 a 9 a 2 + 20 a 3 16 a a = a 2 9a+20 a 16 a 2 a = a 2 9a+20 a 2 (16 a 2 ) = (a5 ) (a4 ) a 2 (4a ) (4+a ) = (a5 )(a4 ) a 2 (a4 )(4+a ) = 5a a 2 (a+4 )
  14. 20 x 2 +7x6 x 4 x 2 25 = 20 x 2 +15x8x6 x 4 x 2 25 x 2 = x[5x(4x+3 ) 2(4x+3 ) ] 4 x 2 25 = x(5x2 ) (4x+3 ) (2x5 ) (2x+5 ) = x (2x+5 )(4x+3 ) (2x5 )(2x+5 ) = x(4x+3 ) 52x
  15. 1+ 1 x1 1+ 1 x 2 1 = x 1 + 1 x1 x 2 1 + 1 x 2 1 = x ( x 2 1 ) x 2 (x1 ) = (x+1 )(x1 ) x (x1 ) = x+1 x
  16. a ab a+b a+ ab ab = a(a+b ) ab a+b a(ab ) +ab ab = a[(a+b ) b ] a+b a[(ab ) +b ] ab = a (a+ b b ) a+b a (a b + b ) ab = a (ab ) a (a+b ) = ab a+b
  17. x1 5 x+3 x+5 35 x+3 = (x1 ) (x+3 ) 5 x+3 (x+5 ) (x+3 ) 35 x+3 = x 2 +2x35 x 2 +8x+1535 = x 2 +2x8 x 2 +8x20 = (x+4 )(x2 ) (x+10 )(x2 ) = x+4 x+10
  18. a+2 7a+9 a+3 a4+ 5a11 a+1 = (a+2 ) (a+3 ) (7a+9 ) a+3 (a4 ) (a+1 ) +5a11 a+1 = a 2 +5a+67a9 a+3 a 2 3a4+5a11 a+1 = a 2 2a3 a+3 a 2 +2a15 a+1 = (a+1 ) ( a 2 2a3 ) (a+3 ) ( a 2 +2a15 ) = (a+1 )(a3 )(a+1 ) (a+3 ) (a+5 )(a3 ) = (a+1 ) 2 (a+3 ) (a+5 )

Ejercicio 136

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 136
Simplificar:
  1. 3x 4y × 8y 9x ÷ z 2 3 x 2 = 3 x 4 y × y 9 x × 3 x 2 z 2 = 2 x 2 z 2
  2. 5a b ÷ ( 2a b 2 × 5x 4 a 2 ) = 5a b ÷ ( 2 a b 2 × 5x a 2 ) = 5a b ÷ 5x 2a b 2 = 5 a b × 2a b 2 5 x = 2 a 2 b x
  3. a+1 a1 × 3a3 2a+2 ÷ a 2 +a a 2 +a+2 = a+1 a1 × 3a3 2a+2 × a 2 +a+2 a 2 +a = a+1 a1 × 3 (a1 ) 2 (a+1 ) × a 2 +a+2 a(a+1 ) = 3( a 2 +a+2 ) 2a(a+1 )
  4. 64 a 2 81 b 2 x 2 81 × (x9 ) 2 8a9b ÷ 8 a 2 +9ab (x+9 ) 2 = 64 a 2 81 b 2 x 2 81 × (x9 ) 2 8a9b ÷ (x+9 ) 2 8 a 2 +9ab = (8a+9b ) (8a9b ) (x+9 ) (x9 ) × (x9 ) 2 8a9b × (x+9 ) 2 a (8a+9b ) = x 2 81 a
  5. x 2 x12 x 2 49 × x 2 x56 x 2 +x20 ÷ x 2 5x24 x+5 = x 2 x12 x 2 49 × x 2 x56 x 2 +x20 × x+5 x 2 5x24 = (x4 ) (x+3 ) (x7 )(x+7 ) × (x8 ) (x+7 ) (x+5 ) (x4 ) × x+5 (x8 ) (x+3 ) = 1 x7
  6. a 2 8a+7 a 2 11a+30 × a 2 36 a 2 1 ÷ a 2 a42 a 2 4a5 = a 2 8a+7 a 2 11a+30 × a 2 36 a 2 1 × a 2 4a5 a 2 a42 = (a7 ) (a1 ) (a6 ) (a5 ) × (a+6 ) (a6 ) (a+1 ) (a1 ) × (a5 ) (a+1 ) (a7 ) (a+6 ) =1
  7. x 4 27x x 2 +7x30 × x 2 +20x+100 x 3 +3 x 2 +9x ÷ x 2 100 x3 = x 4 27x x 2 +7x30 × x 2 +20x+100 x 3 +3 x 2 +9x × x3 x 2 100 = x ( x 3 27 ) (x+10 ) (x3 ) × (x+10 ) 2 x ( x 2 +3x+9 ) × x3 (x10 )(x+10 ) = (x3 )( x 2 +3x+9 ) (x10 )( x 2 +3x+9 ) = x3 x10
  8. a 2 +1 3a6 ÷ ( a 3 +a 6a12 × 4x+8 x3 ) = a 2 +1 3(a2 ) ÷ [ a( a 2 +1 ) (a2 ) × (x+2 ) x3 ] = a 2 +1 3(a2 ) ÷ 2a( a 2 +1 ) (x+2 ) 3(a2 ) (x3 ) = a 2 +1 3 (a2 ) × 3 (a2 )(x3 ) 2a ( a 2 +1 )(x+2 ) = x3 2a(x+2 )
  9. 8 x 2 10x3 6 x 2 +13x+6 × 4 x 2 9 3 x 2 +2x ÷ 8 x 2 +14x+3 9 x 2 +12x+4 = 8 x 2 10x3 6 x 2 +13x+6 × 4 x 2 9 3 x 2 +2x × 9 x 2 +12x+4 8 x 2 +14x+3 = 8 x 2 +2x12x3 6 x 2 +9x+4x+6 × (2x3 ) (2x+3 ) x (3x+2 ) × (3x+2 ) 2 8 x 2 +2x+12x+3 = 2x(4x+1 ) 3(4x+1 ) 3x(2x+3 ) +2(2x+3 ) × (2x3 ) (2x+3 ) x × (3x+2 ) 2x(4x+1 ) +3(4x+1 ) = (2x3 )(4x+1 ) (3x+2 )(2x+3 ) × (2x3 )(2x+3 ) x × (3x+2 ) (2x+3 ) (4x+1 ) = (2x3 ) 2 x(2x+3 )
  10. (a+b ) 2 c 2 (ab ) 2 c 2 × (a+c ) 2 b 2 a 2 +abac ÷ a+b+c a 2 = (a+b ) 2 c 2 (ab ) 2 c 2 × (a+c ) 2 b 2 a 2 +abac × a 2 a+b+c = [(a+b ) +c ] [(a+b ) c ] [(ab ) c ] [(ab ) +c ] × [(a+c ) +b ] [(a+c ) b ] a (a+b+c ) × a 2 a+b+c = (a+b+c )(a+bc ) (abc )(ab+c ) × (a+b+c ) (ab+c ) a+b+c × a a+b+c = a(a+bc ) abc
  11. a 2 5a b+ b 2 ÷ ( a 2 +6a55 b 2 1 × ax+3a a b 2 +11 b 2 ) = a(a5 ) b(1+b ) ÷ [ (a+11 )(a5 ) (b+1 ) (b1 ) × a(x+3 ) b 2 (a+11 ) ] = a(a5 ) b(1+b ) ÷ [ a(x+3 ) (a5 ) b 2 (b+1 ) (b1 ) ] = a (a5 ) b (1+b ) × b 2 (b+1 )(b1 ) a (x+3 )(a5 ) = b(b1 ) x+3
  12. m 3 +6 m 2 n+9m n 2 2 m 2 n+7m n 2 +3 n 3 × 4 m 2 n 2 8 m 2 2mn n 2 ÷ m 3 +27 n 3 16 m 2 +8mn+ n 2 = m 3 +6 m 2 n+9m n 2 2 m 2 n+7m n 2 +3 n 3 × 4 m 2 n 2 8 m 2 2mn n 2 × 16 m 2 +8mn+ n 2 m 3 +27 n 3 = m( m 2 +6mn+9 n 2 ) n(2 m 2 +7mn+3 n 2 ) × (2mn ) (2m+n ) 8 m 2 4mn+2mn n 2 × (4m+n ) 2 (m+3n ) ( m 2 3mn+9 n 2 ) = m (m+3n ) 2 n(2 m 2 +mn+6mn+3 n 2 ) × (2mn ) (2m+n ) 4m(2mn ) +n(2mn ) × (4m+n ) 2 (m+3n )( m 2 3mn+9 n 2 ) = m(m+3n ) n[m(2m+n ) +3n(2m+n ) ] × (2mn )(2m+n ) (4m+n ) (2mn ) × (4m+n ) 2 m 2 3mn+9 n 2 = m (m+3n ) n (m+3n ) (2m+n ) × 2m+n × 4m+n m 2 3mn+9 n 2 = m(4m+n ) n( m 2 3mn+9 n 2 )
  13. ( a 2 ax ) 2 a 2 + x 2 × 1 a 3 + a 2 x ÷ ( a 3 a 2 x a 2 +2ax+ x 2 × a 2 x 2 a 3 +a x 2 ) = [a(ax ) ] 2 a 2 + x 2 × 1 a 2 (a+x ) ÷ [ a 2 (ax ) (a+x ) 2 × (ax )(a+x ) a ( a 2 + x 2 ) ] = a 2 (ax ) 2 a 2 + x 2 × 1 a 2 (a+x ) ÷ a (ax ) 2 (a+x ) ( a 2 + x 2 ) = (ax ) 2 a 2 + x 2 × 1 (a+x ) × (a+x ) ( a 2 + x 2 ) a (ax ) 2 = 1 a
  14. ( a 2 3a ) 2 9 a 2 × 27 a 3 (a+3 ) 2 3a ÷ a 4 9 a 2 ( a 2 +3a ) 2 = ( a 2 3a ) 2 9 a 2 × 27 a 3 (a+3 ) 2 3a × ( a 2 +3a ) 2 a 4 9 a 2 = [a(a3 ) ] 2 (3a )(3+a ) × (3a )(9+3a+ a 2 ) a 2 +6a+93a × [a(a+3 ) ] 2 a 2 ( a 2 9 ) = a 2 (a3 ) 2 3+a × 9+3a+ a 2 a 2 +3a+9 × a 2 (a+3 ) 2 a 2 (a3 ) (a+3 ) = a 2 (a3 )

Ejercicio 135

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 135
Simplificar:
  1. (1+ a a+b ) ÷ (1+ 2a b ) = 1+ a a+b 1+ 2a b = a+b+a a+b b+2a b = 2a+b a+b 2a+b b = b a+b
  2. (x 2 x+1 ) ÷ (x x x+1 ) = x 2 x+1 x x x+1 = x(x+1 ) 2 x+1 x(x+1 ) x x+1 = x 2 +x2 x 2 + x x = x 2 +x2 x 2
  3. (1a+ a 2 1+a ) ÷ (1+ 2 a 2 1 ) = 1a+ a 2 1+a 1+ 2 a 2 1 = (1a ) (1+a ) + a 2 1+a a 2 1+2 a 2 1 = 1 a 2 + a 2 1+a a 2 +1 a 2 1 = a 2 1 (1+a ) ( a 2 +1 ) = (a+1 )(a1 ) (1+a )( a 2 +1 ) = a1 a 2 +1
  4. (x+ 2 x+3 ) ÷ (x+ 3 x+4 ) = x+ 2 x+3 x+ 3 x+4 = x(x+3 ) +2 x+3 x(x+4 ) +3 x+4 = x 2 +3x+2 x+3 x 2 +4x+3 x+4 = (x+4 ) ( x 2 +3x+2 ) (x+3 ) ( x 2 +4x+3 ) = (x+4 ) (x+2 )(x+1 ) (x+3 ) (x+3 )(x+1 ) = (x+4 ) (x+2 ) (x+3 ) 2
  5. (a+b+ b 2 ab ) ÷ (1 b a+b ) =(a+b+ b 2 ab ) × ( 1 1 b a+b ) =[ (a+b ) (ab ) + b 2 ab ] × ( 1 a+ b b a+b ) =( a 2 b 2 + b 2 ab ) × ( 1 a+ b b a+b ) =( a 2 ab ) × ( a+b a ) = a(a+b ) ab
  6. (1 1 x 3 +2 ) ÷ (x+ 1 x1 ) =(1 1 x 3 +2 ) × ( 1 x+ 1 x1 ) =( x 3 +21 x 3 +2 ) × [ 1 x(x1 ) +1 x1 ] =( x 3 +1 x 3 +2 ) × [ 1 x 2 x+1 x1 ] = (x+1 )( x 2 x+1 ) x 3 +2 × x1 x 2 x+1 = x 2 1 x 3 +2
  7. (x+ 1 x+2 ) ÷ (1+ 3 x 2 4 ) =(x+ 1 x+2 ) × ( 1 1+ 3 x 2 4 ) =[ x(x+2 ) +1 x+2 ] × ( 1 x 2 4+3 x 2 4 ) = x 2 +2x+1 x+2 × x 2 4 x 2 1 = (x+1 ) 2 x+2 × (x2 )(x+2 ) (x1 )(x+1 ) = (x+1 ) (x2 ) x1
  8. (n 2n1 n 2 +2 ) ÷ ( n 2 +1 n1 n ) =(n 2n1 n 2 +2 ) × ( 1 n 2 +1 n1 n ) =[ n( n 2 +2 ) (2n1 ) n 2 +2 ] × [ 1 n( n 2 +1 ) (n1 ) n ] =[ n 3 + 2n 2n +1 n 2 +2 ] × [ n n 3 + n n +1 ] = n 3 +1 n 2 +2 × n n 3 +1 = n n 2 +2

Ejercicio 134

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 134
Simplificar:
  1. x 2 3 y 2 ÷ 2x y 3 = x 2 3 y 2 2 x y 3 = xy 6
  2. 3 a 2 b 5 x 2 ÷ a 2 b 3 = 3 a 2 b 5 x 2 a 2 b = 3 5 b 2 x 2
  3. 5 m 2 7 n 3 ÷ 10 m 4 14a n 4 = 5 m 2 7 n 3 m a n 4 = an m 2
  4. 6 a 2 x 3 ÷ a 2 x 5 = 6 a 2 x a 2 x 5 =30 x 2
  5. 15 m 2 19a x 3 ÷ 20 y 2 38 a 3 x 4 = m 2 19 a x 3 y 2 a x 4 = 3 a 2 m 2 x 2 y 2
  6. 11 x 2 y 3 7 m 2 ÷ 22 y 4 = 11 x 2 y 3 7 m 2 y 4 = x 2 14 m 2 y
  7. x1 3 ÷ 2x2 6 = x1 3 2 (x1 ) =1
  8. 3 a 2 a 2 +6ab+9 b 2 ÷ 5 a 3 a 2 b+3a b 2 = 3 a 2 a 2 +6ab+9 b 2 5 a 3 a 2 b+3a b 2 = 3( a 2 b+3a b 2 ) 5( a 2 +6ab+9 b 2 ) = 3ab (a+3b ) 5 (a+3b ) 2 = 3ab 5(a+3b )
  9. x 3 x 2 x 2 +6x ÷ 5 x 2 5x 2x+6 = x ( x 2 1 ) 2 x (x+3 ) 5x(x1 ) 2 (x+3 ) = ( x 2 1 ) 5x(x1 ) = (x1 )(x+1 ) 5x (x1 ) = x+1 5x
  10. 1 a 2 a30 ÷ 2 a 2 +a42 = 1 a 2 a30 × a 2 +a42 2 = 1 (a6 )(a+5 ) × (a+7 )(a6 ) 2 = a+7 2(a+5 )
  11. 20 x 2 30x 15 x 3 +15 x 2 ÷ 4x6 x+1 = 20 x 2 30x 15 x 3 +15 x 2 × x+1 4x6 = x (2x3 ) x 2 (x+1 ) × x+1 2 (2x3 ) = 1 3x
  12. a 2 6a+5 a 2 15a+56 ÷ a 2 +2a35 a 2 5a24 = a 2 6a+5 a 2 15a+56 × a 2 5a24 a 2 +2a35 = (a5 )(a1 ) (a8 )(a7 ) × (a8 )(a+3 ) (a+7 )(a5 ) = (a+3 ) (a1 ) a 2 49
  13. 8 x 2 +26x+15 16 x 2 9 ÷ 6 x 2 +13x5 9 x 2 1 = 8 x 2 +26x+15 16 x 2 9 × 9 x 2 1 6 x 2 +13x5 = 8 x 2 +6x+20x+15 (4x3 ) (4x+3 ) × (3x1 ) (3x+1 ) 6 x 2 2x+15x5 = 2x(4x+3 ) +5(4x+3 ) (4x3 ) (4x+3 ) × (3x1 ) (3x+1 ) 2x(3x1 ) +5(3x1 ) = (2x+5 ) (4x+3 ) (4x3 )(4x+3 ) × (3x1 )(3x+1 ) (2x+5 ) (3x1 ) = 3x+1 4x3
  14. x 3 121x x 2 49 ÷ x 2 11x x+7 = x 3 121x x 2 49 × x+7 x 2 11x = x ( x 2 121 ) (x7 )(x+7 ) × x+7 x (x11 ) = (x+11 )(x11 ) x7 × 1 x11 = x+11 x7
  15. a x 2 +5 4 a 2 1 ÷ a 3 x 2 +5 a 2 2a1 = a x 2 +5 4 a 2 1 × 2a1 a 3 x 2 +5 a 2 = a x 2 +5 (2a+1 )(2a1 ) × 2a1 a 2 (a x 2 +5 ) = 1 a 2 (2a+1 )
  16. a 4 1 a 3 + a 2 ÷ a 4 +4 a 2 +3 3 a 3 +9a = a 4 1 a 3 + a 2 × 3 a 3 +9a a 4 +4 a 2 +3 = ( a 2 +1 )( a 2 1 ) a 2 (a+1 ) × 3 a ( a 2 +3 ) ( a 2 +3 ) ( a 2 +1 ) = 3 (a+1 )(a1 ) a (a+1 ) = 3(a1 ) a
  17. x 3 +125 x 2 64 ÷ x 3 5 x 2 +25x x 2 +x56 = x 3 +125 x 2 64 × x 2 +x56 x 3 5 x 2 +25x = (x+5 )( x 2 5x+25 ) (x+8 )(x+8 ) × (x+8 )(x7 ) x ( x 2 5x+25 ) = (x+5 ) (x7 ) x(x+8 )
  18. 16 x 2 24xy+9 y 2 16x12y ÷ 64 x 3 27 y 3 32 x 2 +24xy+18 y 2 = 16 x 2 24xy+9 y 2 16x12y × 32 x 2 +24xy+18 y 2 64 x 3 27 y 3 = (4x3y ) 2 4 (4x3y ) × 4 (16 x 2 +12y+9 y 2 ) (4x3y ) (16 x 2 +12y+9 y 2 ) =1
  19. a 2 6a a 3 +3 a 2 ÷ a 2 +3a54 a 2 +9a = a 2 6a a 3 +3 a 2 × a 2 +9a a 2 +3a54 = a (a6 ) a 2 (a+3 ) × a (a+9 ) a 2 6a+9a54 = a6 a+3 × a+9 a(a6 ) +9(a6 ) = a6 a+3 × a+9 (a+9 ) (a6 ) = 1 a+3
  20. 15 x 2 +7x2 25 x 3 x ÷ 6 x 2 +13x+6 25 x 2 +10x+1 = 15 x 2 +7x2 25 x 3 x × 25 x 2 +10x+1 6 x 2 +13x+6 = 15 x 2 +10x3x2 x(25 x 2 1 ) × (5x+1 ) 2 6 x 2 +9x+4x+6 = 5x(3x+2 ) (3x+2 ) x (5x+1 )(5x1 ) × (5x+1 ) 2 3x(2x+3 ) +2(2x+3 ) = (5x1 ) (3x+2 ) x (5x1 ) × 5x+1 (3x+2 )(2x+3 ) = 5x+1 x(2x+3 )
  21. x 3 1 2 x 2 2x+2 ÷ 7 x 2 +7x+7 7 x 3 +7 = x 3 1 2 x 2 2x+2 × 7 x 3 +7 7 x 2 +7x+7 = (x1 )( x 2 +x+1 ) 2( x 2 x+1 ) × 7 ( x 3 +1 ) 7 ( x 2 +x+1 ) = (x1 )( x 2 +x+1 ) 2 ( x 2 x+1 ) × (x+1 )( x 2 x+1 ) = x 2 1 2
  22. 2mx2my+nxny 3x3y ÷ 8m+4n = 2mx2my+nxny 3x3y × 1 8m+4n = 2m(xy ) +n(xy ) 3(xy ) × 1 4(2m+n ) = (2m+n ) (xy ) 3 (xy ) × 1 4 (2m+n ) = 1 12
  23. x 2 6x+9 4 x 2 1 ÷ x 2 +5x24 2 x 2 +17x+8 = x 2 6x+9 4 x 2 1 × 2 x 2 +17x+8 x 2 +5x24 = (x3 ) 2 (2x1 ) (2x+1 ) × 2 x 2 +x+16x+8 x 2 3x+8x24 = (x3 ) 2 (2x1 ) (2x+1 ) × x(2x+1 ) +8(2x+1 ) x(x3 ) +8(x3 ) = (x3 ) 2 (2x1 )(2x+1 ) × (x+8 ) (2x+1 ) (x+8 ) (x3 ) = x3 2x1
  24. 2 a 2 +7ab15 b 2 a 3 +4 a 2 b ÷ a 2 3ab40 b 2 a 2 4ab32 b 2 = 2 a 2 +7ab15 b 2 a 3 +4 a 2 b × a 2 4ab32 b 2 a 2 3ab40 b 2 = 2 a 2 +10ab3ab15 b 2 a 2 (a+4b ) × a 2 8ab+4ab32 b 2 a 2 8ab+5ab40 b 2 = 2a(a+5b ) 3b(a+5b ) a 2 (a+4b ) × a(a8b ) +4b(a8b ) a(a8b ) +5b(a8b ) = (2a3b )(a+5b ) a 2 (a+4b ) × (a+4b ) (a8b ) (a+5b ) (a8b ) = 2a3b a 2

Ejercicio 133

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 133
Simplificar:
  1. (a+ a b ) (a a b+1 ) =( ab+a b ) [ a(b+1 ) a b+1 ] = a (b+1 ) b ( ab+ a a b+1 ) = a 2 b b = a 2
  2. (x 2 x+1 ) (x+ 1 x+2 ) =[ x(x+1 ) 2 x+1 ] [ x(x+2 ) +1 x+2 ] =[ x 2 +x2 x+1 ] [ x 2 +2x+1 x+2 ] = (x+2 )(x1 ) x+1 (x+1 ) 2 x+2 = x 2 1
  3. (1 x a+x ) (1+ x a ) =[ a+ x x a+x ] [ a+x a ] = a a =1
  4. (a+ ab ab ) (1 b 2 a 2 ) =[ a(ab ) +ab ab ] [ a 2 b 2 a 2 ] =[ a 2 ab + ab ab ] (a+b )(ab ) a 2 = a 2 (a+b ) a 2 =a+b
  5. (x+2 12 x+1 ) (x2+ 103x x+5 ) =[ (x+1 ) (x+2 ) 12 x+1 ] [ (x2 ) (x+5 ) +103x x+5 ] =[ x 2 +3x+212 x+1 ] [ x 2 + 3x 10 + 10 3x x+5 ] =[ x 2 +3x10 x+1 ] ( x 2 x+5 ) = (x+5 )(x2 ) x+1 ( x 2 x+5 ) = x 2 (x2 ) x+1
  6. (1+ x y ) (x x 2 x+y ) =( y+x y ) [ x(x+y ) x 2 x+y ] =( 1 y ) [ x 2 +xy x 2 x+y ] = x y y =x
  7. (a+x ax+ x 2 a+2x ) (1+ x a+x ) =[ (a+x ) (a+2x ) (ax+ x 2 ) a+2x ] [ a+x+x a+x ] =[ a 2 +2ax+ ax +2 x 2 ax x 2 a+2x ] [ a+2x a+x ] =( a 2 +2ax+ x 2 ) ( 1 a+x ) = (a+x ) 2 ( 1 a+x ) =a+x
  8. (x x 3 6x x 2 25 ) (x+1 8 x+3 ) =[ x( x 2 25 ) ( x 3 6x ) x 2 25 ] [ (x+1 ) (x+3 ) 8 x+3 ] =[ x 3 25x x 3 +6x x 2 25 ] [ x 2 +4x+38 x+3 ] =( 19x x 2 25 ) ( x 2 +4x5 x+3 ) =[ 19x (x5 )(x+5 ) ] [ (x+5 )(x1 ) x+3 ] = 19x19 x 2 (x5 ) (x+3 )
  9. (m mn m+n ) (1+ n 3 m 3 ) =[ m(m+n ) mn m+n ] ( m 3 + n 3 m 3 ) =[ m 2 + mn mn m+n ] [ (m+n )( m 2 mn+ n 2 ) m 3 ] = m 2 (m+n )( m 2 mn+ n 2 ) m 3 = m 2 mn+ n 2 m
  10. (a+2x 14 x 2 2a+x ) (ax+ a 2 +5 x 2 a+4x ) =[ (a+2x ) (2a+x ) 14 x 2 2a+x ] [ (ax ) (a+4x ) + a 2 +5 x 2 a+4x ] =[ 2 a 2 +ax+4ax+2 x 2 14 x 2 2a+x ] [ a 2 +4axax4 x 2 + a 2 +5 x 2 a+4x ] =[ 2 a 2 +5ax12 x 2 2a+x ] [ 2 a 2 +3ax+ x 2 a+4x ] =[ 2 a 2 3ax+8ax12 x 2 2a+x ] [ 2 a 2 +2ax+ax+ x 2 a+4x ] =[ a(2a3x ) +4x(2a3x ) 2a+x ] [ 2a(a+x ) +x(a+x ) a+4x ] = (a+4x )(2a3x ) 2a+x × (2a+x )(a+x ) a+4x =(2a3x ) (a+x )
  11. (1+ a b ) (1 b a ) (1+ b 2 a 2 b 2 ) =( b+a b ) ( ab a ) [ a 2 b 2 + b 2 (ab ) (a+b ) ] = a 2 a b = a b
  12. (2+ 2 x+1 ) (3 6 x+2 ) (1+ 1 x ) =[ 2(x+1 ) +2 x+1 ] [ 3(x+2 ) 6 x+2 ] ( x+1 x ) =(2x+4 ) [ 3x+ 6 6 x+2 ] ( 1 x ) =2 (x+2 )( 3 x x+2 ) ( 1 x ) =6

Ejercicio 132

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 132
Simplificar:
  1. 2 a 2 3b × 6 b 2 4a = 2 a 2 3b × 6 b 2 4 a =ab
  2. x 2 y 5 × 10 a 3 3 m 2 × 9m x 3 = x 2 y 5 × a 3 3 m 2 × m x 3 = 6 a 3 y mx
  3. 5 x 2 7 y 3 × 4 y 2 7 m 3 × 14m 5 x 4 = 5 x 2 7 y 3 × 4 y 2 7 m × m 5 x = 8 7 m 2 x 2 y
  4. 5 a × 2a b 2 × 3b 10 = 5 a × 2 a b 2 × 3 b 10 = 3 b
  5. 2 x 3 15 a 3 × 3 a 2 y × 5 x 2 7x y 2 = 2 x 3 15 a 3 × 3 a 2 y × 5 x 2 7 x y 2 = 2 x 4 7a y 3
  6. 7a 6 m 2 × 3m 10 n 2 × 5 n 4 14ax = 7 a m 2 × 3 m n 2 × 5 n a x = n 2 8mx
  7. 2 x 2 +x 6 × 8 4x+2 = x (2x+1 ) × 2 (2x+1 ) = 2x 3
  8. 5x+25 14 × 7x+7 10x+50 = 5 (x+5 ) × 7 (x+1 ) (x+5 ) = x+1 4
  9. m+n mn n 2 × n 2 m 2 n 2 = (m+n ) n (mn ) × n 2 (mn )(m+n ) = n (mn ) 2
  10. xy2 y 2 x 2 +xy × x 2 +2xy+ y 2 x 2 2xy = y (x2y ) x (x+y ) × (x+y ) 2 x (x2y ) = y(x+y ) x 2
  11. x 2 4xy+4 y 2 x 2 +2xy × x 2 x 2 4 y 2 = (x2y ) 2 x (x+2y ) × x 2 (x+2y )(x2y ) = x(x2y ) (x+2y ) 2
  12. 2 x 2 +2x 2 x 2 × x 2 3x x 2 2x3 = 2 x (x+1 ) 2 x 2 × x (x3 ) (x3 ) (x+1 ) =1
  13. a 2 ab+ab a 2 +2a+1 × 3 6 a 2 6ab = a(ab ) +(ab ) (a+1 ) 2 × 3 6a(ab ) = (ab ) (a+1 ) (a+1 ) 2 × 3 a (ab ) = 1 2(a+1 )
  14. (xy ) 3 x 3 1 × x 2 +x+1 (xy ) 2 = (xy ) 3 (x1 )( x 2 +x+1 ) × x 2 +x+1 (xy ) 2 = xy x1
  15. 2a2 2 a 2 50 × a 2 4a5 3a+3 = 2 (a1 ) 2 ( a 2 25 ) × (a5 )(a+1 ) 3 (a+1 ) = a1 (a+5 )(a5 ) × a5 3 = a1 3(a+5 )
  16. 2 x 2 3x2 6x+3 × 3x+6 x 2 4 = 2 x 2 4x+x2 3(2x+1 ) × 3 (x+2 ) (x2 )(x+2 ) = 2x(x2 ) +x2 3(2x+1 ) × 3 x2 = (2x+1 ) (x2 ) 3 (2x+1 ) × 3 (x2 ) =1
  17. y 2 +9y+18 y5 × 5y25 5y+15 = (y+6 )(y+3 ) y5 × 5 (y5 ) 5 (y+3 ) =y+6
  18. x 3 +2 x 2 3x 4 x 2 +8x+3 × 2 x 2 +3x x 2 x = x( x 2 +2x3 ) 4 x 2 +2x+6x+3 × x (2x+3 ) x (x1 ) = x(x+3 )(x1 ) 2x(2x+1 ) +3(2x+1 ) × 2x+3 x1 = x(x+3 ) (2x+3 )(2x+1 ) × (2x+3 ) = x(x+3 ) 2x+1
  19. x 3 27 a 3 1 × a 2 +a+1 x 2 +3x+9 = (x3 )( x 2 +3x+9 ) (a1 )( a 2 +a+1 ) × a 2 +a+1 x 2 +3x+9 = x3 a1
  20. a 2 +4ab+4 b 2 3 × 2a+4b (a+2b ) 3 = (a+2b ) 2 3 × 2 (a+2b ) (a+2b ) 3 = 2 3
  21. 1x a+1 × a 2 +a x x 2 × x 2 a = 1x a+1 × a (a+1 ) x (1x ) × x 2 a =x
  22. x 2 +2x x 2 16 × x 2 2x8 x 3 + x 2 × x 2 +4x x 2 +4x+4 = x (x+2 ) (x4 )(x+4 ) × x 2 4x+2x8 x 2 (x+1 ) × x (x+4 ) (x+2 ) 2 = 1 x4 × x(x4 ) +2(x4 ) x+1 × 1 x+2 = 1 x4 × (x4 ) (x+2 ) x+1 × 1 x+2 = 1 x+1
  23. (m+n ) 2 x 2 (m+x ) 2 n 2 × (mn ) 2 x 2 m 2 +mnmx = [(m+n ) x ] [(m+n ) +x ] [(m+x ) n ] [(m+x ) +n ] × [(mn ) x ] [(mn ) +x ] m(m+nx ) = [m+nx ] [m+n+x ] [m+xn ] [m+x+n ] × [mnx ][mn+x ] m (m+nx ) = mnx m
  24. 2 a 3 +2a b 2 2a x 2 2ax × x 3 x a 2 x+ b 2 x × x x+1 = 2a ( a 2 + b 2 ) 2a x (x1 ) × x ( x 2 1 ) x ( a 2 + b 2 ) × x x+1 = 1 x1 × (x1 ) (x+1 ) × 1 x+1 =1
  25. a 2 5a+6 3a15 × 6a a 2 a30 × a 2 25 2a4 = (a3 )(a2 ) 3 (a5 ) × 6 a (a6 )(a+5 ) × (a+5 ) (a5 ) 2 (a2 ) = a(a3 ) a6
  26. x 2 3xy10 y 2 x 2 2xy8 y 2 × x 2 16 y 2 x 2 +4xy × x 2 6xy x+2y = (x5y )(x+2y ) (x4y ) (x+2y ) × (x4y ) (x+4y ) x (x+4y ) × x (x6y ) x+2y = (x5y ) (x6y ) x+2y
  27. x 2 +4ax+4 a 2 3ax6 a 2 × 2ax4 a 2 ax+a × 6a+6x x 2 +3ax+2 a 2 = (x+2a ) 2 3 a (x2a ) × 2 a (x2a ) a (x+1 ) × (a+x ) (x+2a ) (x+a ) = 4(x+2a ) a(x+1 )
  28. a 2 81 2 a 2 +10a × a+11 a 2 36 × 2a12 2a+18 × a 3 +5 a 2 2a+22 = (a+9 )(a9 ) 2 a (a+5 ) × a+11 (a6 )(a+6 ) × 2 (a6 ) 2 (a+9 ) × a 2 (a+5 ) 2 (a+11 ) = a(a9 ) 4(a+6 )
  29. a 2 +7a+10 a 2 6a7 × a 2 3a4 a 2 +2a15 × a 3 2 a 2 3a a 2 2a8 = (a+5 ) (a+2 ) (a7 )(a+1 ) × (a4 ) (a+1 ) (a+5 )(a3 ) × a( a 2 2a3 ) (a4 ) (a+2 ) = 1 a7 × 1 a3 × a (a3 )(a+1 ) = a(a+1 ) a7
  30. x 4 +27x x 3 x 2 +x × x 4 +x x 4 3 x 3 +9 x 2 × 1 x (x+3 ) 2 × x 2 x3 = x ( x 3 +27 ) x ( x 2 x+1 ) × x ( x 3 +1 ) x 2 ( x 2 3x+9 ) × 1 x (x+3 ) 2 × x 2 x3 = (x+3 ) ( x 2 3x+9 ) x 2 x+1 × (x+1 )( x 2 x+1 ) x 2 3x+9 × 1 (x+3 ) 2 × 1 x3 = x+1 (x+3 ) (x3 ) = x+1 x 2 9

Ejercicio 131

CAPITULO XIV

Operaciones con Fracciones
Ejercicio 131
Simplificar:
  1. 1 mn + m n 2 m 2 = 1 mn m m 2 n 2 = 1 mn m (mn ) (m+n ) = 1 mn [1 m m+n ] = 1 mn [ m +n m m+n ] = n m 2 n 2
  2. x 2 x 2 xy 2x yx = x 2 x (xy ) + 2x xy = x xy + 2x xy = x+2x xy = 3x xy
  3. 1 2x x 2 + x x 2 4 = 1 x(2x ) x 4 x 2 = 1 x(2x ) x (2x ) (2+x ) = 2+x x 2 x(2x ) (2+x ) = (2x )(1+x ) x (2x )(2+x ) = x+1 x(x+2 )
  4. a+b a 2 ab + a b 2 a 2 = a+b a(ab ) a a 2 b 2 = a+b a(ab ) a (a+b ) (ab ) = 1 ab [ a+b a a a+b ] = 1 ab [ (a+b ) 2 a 2 a(a+b ) ] = 1 ab [ a 2 +2ab+ b 2 a 2 a(a+b ) ] = 2ab+ b 2 a( a 2 b 2 )
  5. x4 x 2 2x3 x 62x = x4 (x3 ) (x+1 ) + x 2x6 = x4 (x3 ) (x+1 ) + x 2(x3 ) = 1 x3 [ x4 x+1 + x 2 ] = 1 x3 [ 2(x4 ) +x(x+1 ) 2(x+1 ) ] = 1 x3 [ 2x8+ x 2 +x 2(x+1 ) ] = 1 x3 [ x 2 +3x8 2(x+1 ) ] = x 2 +3x8 2(x+1 ) (x3 )
  6. 1 x 2 +2x8 + 1 (2x ) (x+3 ) = 1 (x+4 ) (x2 ) 1 (x2 ) (x+3 ) = 1 x2 [ 1 x+4 1 x+3 ] = 1 x2 [ x+3(x+4 ) (x+4 ) (x+3 ) ] = 1 x2 [ x +3 x 4 (x+4 ) (x+3 ) ] = 1 (x2 ) (x+4 ) (x+3 )
  7. 1 2x+2 + 2 1x + 7 4x4 = 1 2(x+1 ) 2 x1 + 7 4(x1 ) = 2(x1 ) 8(x+1 ) +7(x+1 ) 4(x1 ) (x+1 ) = 2x28x8+7x+7 4( x 2 1 ) = x3 4( x 2 1 )
  8. 2a a+3 + 3a a3 + 2a 9 a 2 = 2a a+3 + 3a a3 2a a 2 9 =a[ 2 a+3 + 3 a3 2 (a+3 ) (a3 ) ] =a[ 2(a3 ) +3(a+3 ) 2 (a+3 ) (a3 ) ] =a[ 2a6+3a+92 a 2 9 ] = a(5a+1 ) a 2 9
  9. x+3y y+x + 3 y 2 x 2 y 2 x yx = x+3y y+x 3 y 2 y 2 x 2 x yx = x+3y y+x 3 y 2 (yx ) (y+x ) x yx = (x+3y ) (yx ) 3 y 2 x(y+x ) (yx ) (y+x ) = xy x 2 + 3 y 2 3xy 3 y 2 xy x 2 y 2 x 2 = 2 x 2 3xy y 2 x 2 = 2 x 2 +3xy x 2 y 2
  10. x x 2 +2x3 + x3 (1x ) (x+2 ) + 1 x+2 = x (x+3 ) (x1 ) x3 (x1 ) (x+2 ) + 1 x+2 = x(x+2 ) (x+3 ) (x3 ) +(x+3 ) (x1 ) (x+3 ) (x1 ) (x+2 ) = x 2 +2x( x 2 9 ) + x 2 +2x3 (x+3 ) (x1 ) (x+2 ) = 2 x 2 +4x3 x 2 +9 (x+3 ) (x1 ) (x+2 ) = x 2 +4x+6 (x+3 ) (x1 ) (x+2 )
  11. 3 2a+2 1 4a4 4 88 a 2 = 3 2(a+1 ) 1 4(a1 ) + 4 8 a 2 8 = 3 2(a+1 ) 1 4(a1 ) + 4 8( a 2 1 ) = 1 2 [ 3 a+1 1 2(a1 ) + 4 4 (a+1 ) (a1 ) ] = 1 2 [ 6(a1 ) (a+1 ) +2 2(a+1 ) (a1 ) ] = 1 2 [ 6a6a1+2 2(a+1 ) (a1 ) ] = 1 2 [ 5a5 2(a+1 ) (a1 ) ] = 1 2 [ 5 (a1 ) 2(a+1 )(a1 ) ] = 5 4(a+1 )
  12. 1 a3 + a+1 (3a ) (a2 ) + 2 (2a ) (1a ) = 1 a3 a+1 (a3 ) (a2 ) + 2 (a2 ) (a1 ) = (a2 ) (a1 ) (a+1 ) (a1 ) +2(a3 ) (a3 ) (a2 ) (a1 ) = a 2 3a+2( a 2 1 ) +2a6 (a3 ) (a2 ) (a1 ) = a 2 3a+2 a 2 +1+2a6 (a3 ) (a2 ) (a1 ) = a3 (a3 ) (a2 ) (a1 ) = a+3 (3a ) (a2 ) (a1 )
  13. 2x x1 + 2 x 3 +2 x 2 1 x 3 + 1 x 2 +x+1 = 2x x1 2 x 2 (x+1 ) x 3 1 + 1 x 2 +x+1 = 2x x1 2 x 2 (x+1 ) (x1 ) ( x 2 +x+1 ) + 1 x 2 +x+1 = 2x( x 2 +x+1 ) 2 x 2 (x+1 ) +x1 (x1 ) ( x 2 +x+1 ) = 2x[( x 2 +x+1 ) x(x+1 ) ] +x1 (x1 ) ( x 2 +x+1 ) = 2x[ x 2 + x +1 x 2 x ] +x1 (x1 ) ( x 2 +x+1 ) = 3x1 x 3 1
  14. x+2 3x1 + x+1 32x + 4 x 2 +6x+3 6 x 2 11x+3 = x+2 3x1 x+1 2x3 + 4 x 2 +6x+3 6 x 2 2x9x+3 = x+2 3x1 x+1 2x3 + 4 x 2 +6x+3 2x(3x1 ) 3(3x1 ) = x+2 3x1 x+1 2x3 + 4 x 2 +6x+3 (2x3 ) (3x1 ) = (2x3 ) (x+2 ) (x+1 ) (3x1 ) +4 x 2 +6x+3 (2x3 ) (3x1 ) = (2 x 2 3x+4x6 ) (3 x 2 x+3x1 ) +4 x 2 +6x+3 (2x3 ) (3x1 ) = 2 x 2 3x +4x63 x 2 +x 3x +1+4 x 2 + 6x +3 (2x3 ) (3x1 ) = 3 x 2 +5x2 (2x3 ) (3x1 ) = 3 x 2 x+6x2 (2x3 ) (3x1 ) = x(3x1 ) +2(3x1 ) (2x3 ) (3x1 ) = (x+2 )(3x1 ) (2x3 )(3x1 ) = x+2 2x3