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CAPITULO XIV

Operaciones con Fracciones
Ejercicio 130
Simplificar:
  1. 2 x3 + 3 x+2 4x7 x 2 x6 = 2 x3 + 3 x+2 4x7 (x3 ) (x+2 ) = 2(x+2 ) +3(x3 ) (4x7 ) (x3 ) (x+2 ) = 2x+4+3x94x+7 (x3 ) (x+2 ) = x+2 (x3 )(x+2 ) = 1 x3
  2. a 3a+6 1 6a+12 + a+12 12a+24 = a 3(a+2 ) 1 6(a+2 ) + a+12 12(a+2 ) = 1 3(a+2 ) [a 1 2 + a+12 4 ] = 1 3(a+2 ) [ 4a2+a+12 4 ] = 1 3(a+2 ) [ 5a+10 4 ] = 1 3 (a+2 ) [ 5 (a+2 ) 4 ] = 5 12
  3. x x 2 +1 + 1 3x 1 x 2 = 3 x 3 +x( x 2 +1 ) 3( x 2 +1 ) 3 x 2 ( x 2 +1 ) = 3 x 3 + x 3 +x3 x 2 3 3 x 2 ( x 2 +1 ) = 4 x 3 3 x 2 +x3 3 x 2 ( x 2 +1 )
  4. a+3 a 2 1 + a1 2a+2 + a4 4a4 = a+3 (a1 ) (a+1 ) + a1 2(a+1 ) + a4 4(a1 ) = 4(a+3 ) +2 (a1 ) 2 +(a+1 ) (a4 ) 4(a1 ) (a+1 ) = 4a+12+2( a 2 2a+1 ) + a 2 3a4 4(a1 ) (a+1 ) = a+8+2 a 2 4a+2+ a 2 4( a 2 1 ) = 3 a 2 3a+10 4( a 2 1 )
  5. ab a 2 +ab + a+b ab a ab+ b 2 = ab a(a+b ) + a+b ab a b(a+b ) = b(ab ) + (a+b ) 2 a 2 ab(a+b ) = ab b 2 + a 2 +2ab+ b 2 a 2 ab(a+b ) = 3 ab ab (a+b ) = 3 a+b
  6. xy x+y x+y xy + 4 x 2 x 2 y 2 = xy x+y x+y xy + 4 x 2 (x+y ) (xy ) = (xy ) 2 (x+y ) 2 +4 x 2 (x+y ) (xy ) = [(xy ) +(x+y ) ] [(xy ) (x+y ) ] +4 x 2 (x+y ) (xy ) = [x y +x+ y ] [ x y x y ] +4 x 2 (x+y ) (xy ) = 2x(2y ) +4 x 2 (x+y ) (xy ) = 4 x 2 4xy (x+y ) (xy ) = 4x (xy ) (x+y )(xy ) = 4x x+y
  7. x a 2 ax + 1 a + 1 x = x a(ax ) + 1 a + 1 x = x 2 +x(ax ) +a(ax ) ax(ax ) = x 2 + ax x 2 + a 2 ax ax(ax ) = a 2 a x(ax ) = a x(ax )
  8. x+1 x 2 x20 x+4 x 2 4x5 + x+5 x 2 +5x+4 = x+1 (x5 ) (x+4 ) x+4 (x5 ) (x+1 ) + x+5 (x+4 ) (x+1 ) = (x+1 ) 2 (x+4 ) 2 +(x5 ) (x+5 ) (x+1 ) (x5 ) (x+4 ) = [(x+1 ) +(x+4 ) ] [(x+1 ) (x+4 ) ] + x 2 25 (x+1 ) (x5 ) (x+4 ) = [x+1+x+4 ] [ x +1 x 4 ] + x 2 25 (x+1 ) (x5 ) (x+4 ) = 3(2x+5 ) + x 2 25 (x+1 ) (x5 ) (x+4 ) = 6x15+ x 2 25 (x+1 ) (x5 ) (x+4 ) = x 2 6x40 (x+1 ) (x5 ) (x+4 ) = (x10 )(x+4 ) (x+1 ) (x5 )(x+4 ) = x10 (x+1 ) (x5 )
  9. 2x+1 12x+8 x 2 6 x 2 +x2 + 2x 16x8 = 2x+1 4(3x+2 ) x 2 6 x 2 3x+4x2 + 2x 8(2x1 ) = 2x+1 4(3x+2 ) x 2 3x(2x1 ) +2(2x1 ) + 2x 8(2x1 ) = 2x+1 4(3x+2 ) x 2 (3x+2 ) (2x1 ) + 2x 8(2x1 ) = 2(2x1 ) (2x+1 ) 8 x 2 +2x(3x+2 ) 8(3x+2 ) (2x1 ) = 2(4 x 2 1 ) 8 x 2 +6 x 2 +4x 8(3x+2 ) (2x1 ) = 8 x 2 22 x 2 +4x 8(3x+2 ) (2x1 ) = 6 x 2 +4x2 8(3x+2 ) (2x1 ) = 2 (3 x 2 +2x1 ) (3x+2 ) (2x1 ) = 3 x 2 +2x1 4(3x+2 ) (2x1 )
  10. 1 ax 1 a 2 +ax + 1 a+x = 1 ax 1 a(a+x ) + 1 a+x = a+ x x +ax ax(a+x ) = a+ax ax(a+x ) = a (1+x ) a x(a+x ) = 1+x x(a+x )
  11. 1 x+y 1 xy + 2y x 2 + y 2 = (xy ) ( x 2 + y 2 ) (x+y ) ( x 2 + y 2 ) +2y(x+y ) (xy ) (x+y ) (xy ) ( x 2 + y 2 ) = x 3 +x y 2 x 2 y y 3 ( x 3 +x y 2 + x 2 y+ y 3 ) +2y( x 2 y 2 ) (x+y ) (xy ) ( x 2 + y 2 ) = x 3 + x y 2 x 2 y y 3 x 3 x y 2 x 2 y y 3 + 2 x 2 y 2 y 3 ( x 2 y 2 ) ( x 2 + y 2 ) = 4 y 3 x 4 y 4 = 4 y 3 y 4 x 4
  12. a1 3a+3 a2 6a6 + a 2 +2a6 9 a 2 9 = a1 3(a+1 ) a2 6(a1 ) + a 2 +2a6 9( a 2 1 ) = 1 3 [ a1 a+1 a2 2(a1 ) + a 2 +2a6 3(a1 ) (a+1 ) ] = 1 3 [ 6 (a1 ) 2 3(a2 ) (a+1 ) +2( a 2 +2a6 ) 6(a1 ) (a+1 ) ] = 1 3 [ 6( a 2 2a+1 ) 3( a 2 a2 ) +2 a 2 +4a12 6(a1 ) (a+1 ) ] = 1 3 [ 6 a 2 12a+ 6 3 a 2 +3a+ 6 +2 a 2 +4a 12 6(a1 ) (a+1 ) ] = 1 3 [ 5 a 2 5a 6(a1 ) (a+1 ) ] = 1 3 [ 5a (a1 ) 6 (a1 )(a+1 ) ] = 5a 18(a+1 )
  13. 1 a 2 +2a24 + 2 a 2 2a8 3 a 2 +8a+12 = 1 (a+6 ) (a4 ) + 2 (a4 ) (a+2 ) 3 (a+6 ) (a+2 ) = a+2+2(a+6 ) 3(a4 ) (a+6 ) (a4 ) (a+2 ) = a +2+ 2a +12 3a +12 (a+6 ) (a4 ) (a+2 ) = 26 (a+6 ) (a4 ) (a+2 )
  14. x+y xy x+2y xy+ y 2 y x 2 +xy = x+y xy x+2y y(x+y ) y x(x+y ) = (x+y ) 2 x(x+2y ) y 2 xy(x+y ) = x 2 + 2xy + y 2 x 2 2xy y 2 xy(x+y ) =0
  15. a 3 a 3 +1 + a+3 a 2 a+1 a1 a+1 = a 3 (a+1 ) ( a 2 a+1 ) + a+3 a 2 a+1 a1 a+1 = a 3 +(a+3 ) (a+1 ) (a1 ) ( a 2 a+1 ) (a+1 ) ( a 2 a+1 ) = a 3 + a 2 +4a+3( a 3 a 2 +a a 2 +a1 ) (a+1 ) ( a 2 a+1 ) = a 3 + a 2 +4a+3 a 3 +2 a 2 2a+1 a 3 +1 = 3 a 2 +2a+4 a 3 +1
  16. 1 x1 + 2x x 2 1 3 x 2 x 3 1 = 1 x1 + 2x (x1 ) (x+1 ) 3 x 2 (x1 ) ( x 2 +x+1 ) = 1 x1 [1+ 2x x+1 3 x 2 x 2 +x+1 ] = 1 x1 [ (x+1 ) ( x 2 +x+1 ) +2x( x 2 +x+1 ) 3 x 2 (x+1 ) (x+1 ) ( x 2 +x+1 ) ] = 1 x1 [ x 3 + x 2 +x+ x 2 +x+1+ 2 x 3 + 2 x 2 +2x 3 x 3 3 x 2 (x+1 ) ( x 2 +x+1 ) ] = 1 x1 [ x 2 +4x+1 (x+1 ) ( x 2 +x+1 ) ] = x 2 +4x+1 (x+1 ) ( x 3 1 )
  17. a+b a 2 ab+ b 2 1 a+b + 3 a 2 a 3 + b 3 = a+b a 2 ab+ b 2 1 a+b + 3 a 2 (a+b ) ( a 2 ab+ b 2 ) = (a+b ) 2 ( a 2 ab+ b 2 ) +3 a 2 (a+b ) ( a 2 ab+ b 2 ) = a 2 +2ab+ b 2 a 2 +ab b 2 +3 a 2 (a+b ) ( a 2 ab+ b 2 ) = 3 a 2 +3ab (a+b ) ( a 2 ab+ b 2 ) = 3a (a+b ) (a+b )( a 2 ab+ b 2 ) = 3a a 2 ab+ b 2
  18. 2 x2 + 2x+3 x 2 +2x+4 6x+12 x 3 8 = 2 x2 + 2x+3 x 2 +2x+4 6x+12 (x2 ) ( x 2 +2x+4 ) = 2( x 2 +2x+4 ) +(x2 ) (2x+3 ) (6x+12 ) (x2 ) ( x 2 +2x+4 ) = 2 x 2 + 4x +8+2 x 2 +3x 4x 66x12 (x2 ) ( x 2 +2x+4 ) = 4 x 2 3x10 (x2 ) ( x 2 +2x+4 ) = 4 x 2 8x+5x10 (x2 ) ( x 2 +2x+4 ) = 4x(x2 ) +5(x2 ) (x2 ) ( x 2 +2x+4 ) = (4x+5 )(x2 ) (x2 )( x 2 +2x+4 ) = 4x+5 x 2 +2x+4
  19. 3x+2 x 2 +3x10 5x+1 x 2 +4x5 + 4x1 x 2 3x+2 = 3x+2 (x+5 ) (x2 ) 5x+1 (x+5 ) (x1 ) + 4x1 (x2 ) (x1 ) = (3x+2 ) (x1 ) (x2 ) (5x+1 ) +(x+5 ) (4x1 ) (x+5 ) (x2 ) (x1 ) = 3 x 2 + 2x 3x+25 x 2 x +10x+2+4 x 2 x +20x5 (x+5 ) (x2 ) (x1 ) = 2 x 2 +27x1 (x+5 ) (x2 ) (x1 )
  20. 1 (n1 ) 2 + 1 n1 1 (n1 ) 3 1 n = 1 (n1 ) 2 + 1 n1 1 (n1 ) 3 1 n = n(n1 ) +n (n1 ) 2 n (n1 ) 3 n (n1 ) 3 = n 2 n+n( n 2 2n+1 ) n( n 3 3 n 2 +3n1 ) n (n1 ) 3 = n 2 2n+ n 3 2 n 2 +n n 3 +3 n 2 3n+1 n (n1 ) 3 = 2 n 2 4n+1 n (n1 ) 3
  21. 1 a 2 +5 a 2 5 ( a 2 +5 ) 2 + a 2 +5 a 4 25 = 1 a 2 +5 a 2 5 ( a 2 +5 ) 2 + a 2 +5 ( a 2 +5 ) ( a 2 5 ) = 1 a 2 +5 [1 a 2 5 a 2 +5 + a 2 +5 a 2 5 ] = 1 a 2 +5 [ ( a 2 +5 ) ( a 2 5 ) ( a 2 5 ) 2 + ( a 2 +5 ) 2 ( a 2 +5 ) ( a 2 5 ) ] = 1 a 2 +5 [ a 4 25 ( a 4 10 a 2 +25 ) + a 4 +10 a 2 + 25 ( a 2 +5 ) ( a 2 5 ) ] = 1 a 2 +5 [ 2 a 4 a 4 +10 a 2 25+10 a 2 ( a 2 +5 ) ( a 2 5 ) ] = 1 a 2 +5 [ a 4 +20 a 2 25 ( a 2 +5 ) ( a 2 5 ) ] = a 4 +20 a 2 25 ( a 2 +5 ) ( a 4 25 )
  22. 1 x 2 9 x 2 x 2 9+6x+ x 2 6x 96x+ x 2 = 1 x 2 (3x ) (3+x ) x 2 (3+x ) 2 6x (3x ) 2 = (1 x 2 ) (3x ) (3+x ) x 2 (3x ) 2 6x (3+x ) 2 (3x ) 2 (3+x ) 2 = (1 x 2 ) (9 x 2 ) x 2 (96x+ x 2 ) 6x(9+6x+ x 2 ) (3x ) 2 (3+x ) 2 = 9 x 2 9 x 2 + x 4 9 x 2 + 6 x 3 x 4 54x36 x 2 6 x 3 (3x ) 2 (3+x ) 2 = 954x55 x 2 (3x ) 2 (3+x ) 2
  23. x 2x+2 x+1 3x3 + x1 6x+6 5 18x18 = x 2(x+1 ) x+1 3(x1 ) + x1 6(x+1 ) 5 18(x1 ) = 9x(x1 ) 6 (x+1 ) 2 +3 (x1 ) 2 5(x+1 ) 18(x1 ) (x+1 ) = 9 x 2 9x6( x 2 +2x+1 ) +3( x 2 2x+1 ) 5x5 18(x1 ) (x+1 ) = 9 x 2 9x6 x 2 12x6+3 x 2 6x+35x5 18(x1 ) (x+1 ) = 6 x 2 32x8 18(x1 ) (x+1 ) = 2 (3 x 2 16x4 ) (x1 ) (x+1 ) = 3 x 2 16x4 9(x1 ) (x+1 )
  24. a+2 2a+2 7a 8 a 2 8 a3 4a4 = a+2 2(a+1 ) 7a 8( a 2 1 ) a3 4(a1 ) = 1 2 [ a+2 a+1 7a 4(a+1 ) (a1 ) a3 2(a1 ) ] = 1 2 [ 4(a+2 ) (a1 ) 7a2(a3 ) (a+1 ) 4(a+1 ) (a1 ) ] = 1 2 [ 4( a 2 +a2 ) 7a2( a 2 2a3 ) 4(a+1 ) (a1 ) ] = 1 2 [ 4 a 2 +4a87a2 a 2 +4a+6 4(a+1 ) (a1 ) ] = 1 2 [ 2 a 2 +a2 4(a+1 ) (a1 ) ] = 2 a 2 +a2 8(a+1 ) (a1 )
  25. a3 20a+10 + 2a+5 40a+20 4a1 60a+30 = a3 10(2a+1 ) + 2a+5 20(2a+1 ) 4a1 30(2a+1 ) = 1 10(2a+1 ) [a3+ 2a+5 2 4a1 3 ] = 1 10(2a+1 ) [ 6a18+3(2a+5 ) 2(4a1 ) 6 ] = 1 10(2a+1 ) [ 6a18+6a+158a+2 6 ] = 1 10(2a+1 ) [ 4a1 6 ] = 4a1 60(2a+1 )
  26. 2 2 x 2 +5x+3 1 2 x 2 x6 + 3 x 2 x2 = 2 2 x 2 +2x+3x+3 1 2 x 2 4x+3x6 + 3 (x2 ) (x+1 ) = 2 2x(x+1 ) +3(x+1 ) 1 2x(x2 ) +3(x2 ) + 3 (x2 ) (x+1 ) = 2 (2x+3 ) (x+1 ) 1 (2x+3 ) (x2 ) + 3 (x2 ) (x+1 ) = 2(x2 ) (x+1 ) +3(2x+3 ) (2x+3 ) (x+1 ) (x2 ) = 2x4x1+6x+9 (2x+3 ) (x+1 ) (x2 ) = 7x+4 (2x+3 ) (x+1 ) (x2 )
  27. a1 a2 a2 a+3 + 1 a1 = (a+3 ) (a1 ) 2 (a2 ) 2 (a1 ) +(a2 ) (a+3 ) (a2 ) (a+3 ) (a1 ) = (a+3 ) ( a 2 2a+1 ) ( a 2 4a+4 ) (a1 ) + a 2 +a6 (a2 ) (a+3 ) (a1 ) = a 3 2 a 2 +a+3 a 2 6a+3( a 3 4 a 2 +4a a 2 +4a4 ) + a 2 +a6 (a2 ) (a+3 ) (a1 ) = a 3 + a 2 5a+3 a 3 +5 a 2 8a+4+ a 2 +a6 (a2 ) (a+3 ) (a1 ) = 7 a 2 12a+1 (a2 ) (a+3 ) (a1 )
  28. 2+3a 23a 23a 2+3a a (23a ) 2 = (2+3a ) 2 (23a ) (23a ) 3 a(2+3a ) (2+3a ) (23a ) 2 = (2+3a ) (49 a 2 ) (836a+54 a 2 27 a 3 ) 2a3 a 2 (2+3a ) (23a ) 2 = 8 18 a 2 +12a 27 a 3 8 +36a54 a 2 + 27 a 3 2a3 a 2 (2+3a ) (23a ) 2 = 46a75 a 2 (2+3a ) (23a ) 2
  29. 1 5+5a + 1 55a 1 10+10 a 2 = 1 5(1+a ) + 1 5(1a ) 1 10(1+ a 2 ) = 1 5 [ 1 1+a + 1 1a 1 2(1+ a 2 ) ] = 1 5 [ 2(1+ a 2 ) (1a ) +2(1+ a 2 ) (1+a ) (1a ) (1+a ) 2(1+ a 2 ) (1a ) (1+a ) ] = 1 5 [ 2(1+ a 2 a a 3 ) +2(1+ a 2 +a+ a 3 ) (1 a 2 ) 2(1+ a 2 ) (1a ) (1+a ) ] = 1 5 [ 2+2 a 2 2a 2 a 3 +2+2 a 2 + 2a + 2 a 3 1+ a 2 2(1+ a 2 ) (1 a 2 ) ] = 1 5 [ 5 a 2 +3 2(1+ a 2 ) (1 a 2 ) ] = 5 a 2 +3 10(1 a 4 )
  30. 1 33x 1 3+3x + x 6+6 x 2 x 22 x 2 = 1 3(1x ) 1 3(1+x ) + x 6(1+ x 2 ) x 2(1 x 2 ) = 2(1+ x 2 ) (1+x ) 2(1+ x 2 ) (1x ) +x(1x ) (1+x ) 3x(1+ x 2 ) 6(1+ x 2 ) (1x ) (1+x ) = 2(1+ x 2 +x+ x 3 ) 2(1+ x 2 x x 3 ) +x(1 x 2 ) 3x3 x 3 6(1+ x 2 ) (1 x 2 ) = 2 + 2 x 2 +2x+ 2 x 3 2 2 x 2 + 2x + 2 x 3 + x x 3 3x 3 x 3 6(1 x 4 ) = 2x 6(1 x 4 ) = 2 x (1 x 4 ) = x 3(1 x 4 )