Ejercicio 129

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CAPITULO XIV

Operaciones con Fracciones
Ejercicio 129

  1. De 1 x4 restar 1 x3
    1 x4 1 x3 = x3(x4 ) (x4 ) (x3 ) = x 3 x +4 (x4 ) (x3 ) = 1 (x4 ) (x3 )
  2. De mn m+n restar m+n mn
    mn m+n m+n mn = (mn ) 2 (m+n ) 2 (m+n ) (mn ) = [(mn ) +(m+n ) ] [(mn ) (m+n ) ] (m+n ) (mn ) = [m n +m+ n ] [ m n m n ] m 2 n 2 = 2m(2n ) m 2 n 2 = 4mn m 2 n 2 = 4mn n 2 m 2
  3. De 1x 1+x restar 1+x 1x
    1x 1+x 1+x 1x = (1x ) 2 (1+x ) 2 (1x ) (1+x ) = [(1x ) +(1+x ) ] [(1x ) (1+x ) ] (1x ) (1+x ) = [1 x +1+ x ] [ 1 x 1 x ] 1 x 2 = 2(2x ) 1 x 2 = 4x 1 x 2 = 4x x 2 1
  4. De a+b a 2 +ab restar ba ab+ b 2
    a+b a 2 +ab ba ab+ b 2 = a+b a(a+b ) ba b(a+b ) = b(a+b ) a(ba ) ab(a+b ) = ab + b 2 ab + a 2 ab(a+b ) = a 2 + b 2 ab(a+b )
  5. De m+n mn restar m 2 + n 2 m 2 n 2
    m+n mn m 2 + n 2 m 2 n 2 = m+n mn m 2 + n 2 (m+n ) (mn ) = (m+n ) 2 ( m 2 + n 2 ) (m+n ) (mn ) = m 2 +2mn+ n 2 m 2 n 2 (m+n ) (mn ) = 2mn m 2 n 2
  6. Restar 1 x x 2 de 1 x+ x 2
    1 x+ x 2 1 x x 2 = 1 x(1+x ) 1 x(1x ) = 1 x [ 1 1+x 1 1x ] = 1 x [ 1x(1+x ) (1+x ) (1x ) ] = 1 x [ 1 x 1 x (1+x ) (1x ) ] = 1 x 2 x (1+x ) (1x ) = 2 1 x 2 = 2 x 2 1
  7. Restar x a 2 x 2 de a+x (ax ) 2
    a+x (ax ) 2 x a 2 x 2 = a+x (ax ) 2 x (ax ) (a+x ) = 1 ax [ a+x (ax ) x a+x ] = 1 ax [ (a+x ) 2 x(ax ) (ax ) (a+x ) ] = 1 ax [ a 2 +2ax+ x 2 ax+ x 2 (ax ) (a+x ) ] = 1 ax [ a 2 +ax+2 x 2 (ax ) (a+x ) ] = a 2 +ax+2 x 2 (ax ) 2 (a+x )
  8. Restar 1 12a+6 de a+1 6a+3
    a+1 6a+3 1 12a+6 = a+1 3(2a+1 ) 1 6(2a+1 ) = 1 3 [ a+1 2a+1 1 2(2a+1 ) ] = 1 3 [ 2(a+1 ) 1 2(2a+1 ) ] = 1 3 [ 2a+21 2(2a+1 ) ] = 2a+1 6(2a+1 )
  9. Restar a+3 a 2 +a12 de a4 a 2 6a+9
    a4 a 2 6a+9 a+3 a 2 +a12 = a4 (a3 ) 2 a+3 (a+4 ) (a3 ) = 1 (a3 ) [ a4 a3 a+3 a+4 ] = 1 (a3 ) [ (a4 ) (a+4 ) (a3 ) (a+3 ) (a+4 ) (a3 ) ] = 1 (a3 ) [ a 2 16( a 2 9 ) (a+4 ) (a3 ) ] = 1 (a3 ) [ a 2 16 a 2 +9 (a+4 ) (a3 ) ] = 1 (a3 ) [ 7 (a+4 ) (a3 ) ] = 7 (a+4 ) (a3 ) 2
  10. Restar b a+3b de a 2 +4ab3 b 2 a 2 9 b 2
    a 2 +4ab3 b 2 a 2 9 b 2 b a+3b = a 2 +4ab3 b 2 (a+3b ) (a3b ) b a+3b = a 2 +4ab3 b 2 b(a3b ) (a+3b ) (a3b ) = a 2 +4ab 3 b 2 ab+ 3 b 2 (a+3b ) (a3b ) = a 2 +3ab (a+3b ) (a3b ) = a (a+3b ) (a+3b )(a3b ) = a a3b
    Simplificar:
  11. x x 2 1 x+1 (x1 ) 2 = x (x1 ) (x+1 ) x+1 (x1 ) 2 = 1 x1 [ x x+1 x+1 x1 ] = 1 x1 [ x(x1 ) (x+1 ) 2 (x1 ) (x+1 ) ] = 1 x1 [ x 2 x( x 2 +2x+1 ) (x1 ) (x+1 ) ] = 1 x1 [ x 2 x x 2 2x1 (x1 ) (x+1 ) ] = 1 x1 [ 3x1 (x1 ) (x+1 ) ] = 3x+1 (x1 ) 2 (x+1 )
  12. 1 a 3 b 3 1 (ab ) 3 = 1 (ab ) ( a 2 +ab+ b 2 ) 1 (ab ) 3 = 1 ab [ 1 a 2 +ab+ b 2 1 (ab ) 2 ] = 1 ab [ (ab ) 2 ( a 2 +ab+ b 2 ) (ab ) 2 ( a 2 +ab+ b 2 ) ] = 1 ab [ a 2 2ab+ b 2 a 2 ab b 2 (ab ) 2 ( a 2 +ab+ b 2 ) ] = 1 ab [ 3ab (ab ) 2 ( a 2 +ab+ b 2 ) ] = 3ab (ab ) 3 ( a 2 +ab+ b 2 )
  13. x+3 6 x 2 +x2 1 4 x 2 4x+1 = x+3 6 x 2 3x+4x2 1 (2x1 ) 2 = x+3 3x(2x1 ) +2(2x1 ) 1 (2x1 ) 2 = x+3 (3x+2 ) (2x1 ) 1 (2x1 ) 2 = 1 2x1 [ x+3 3x+2 1 2x1 ] = 1 2x1 [ (x+3 ) (2x1 ) (3x+2 ) (3x+2 ) (2x1 ) ] = 1 2x1 [ 2 x 2 x+6x33x2 (3x+2 ) (2x1 ) ] = 1 2x1 [ 2 x 2 +2x5 (3x+2 ) (2x1 ) ] = 2 x 2 +2x5 (3x+2 ) (2x1 ) 2
  14. x1 4x+4 x+2 8x8 = x1 4(x+1 ) x+2 8(x1 ) = 1 4 [ x1 x+1 x+2 2(x1 ) ] = 1 4 [ 2 (x1 ) 2 (x+1 ) (x+2 ) 2(x1 ) (x+1 ) ] = 1 4 [ 2( x 2 2x+1 ) ( x 2 +3x+2 ) 2(x1 ) (x+1 ) ] = 1 4 [ 2 x 2 4x+ 2 x 2 3x 2 2(x1 ) (x+1 ) ] = 1 4 [ x 2 7x 2(x1 ) (x+1 ) ] = x 2 7x 8( x 2 1 )
  15. x xy y 2 1 y = x y(xy ) 1 y = 1 y [ x xy 1 ] = 1 y [ x(xy ) xy ] = 1 y [ x x +y xy ] = 1 y [ y xy ] = 1 xy
  16. b a 2 b 2 b a 2 +ab = b (a+b ) (ab ) b a(a+b ) = b a+b [ 1 ab 1 a ] = b a+b [ a(ab ) a(ab ) ] = b a+b [ a a +b a(ab ) ] = b a+b [ b a(ab ) ] = b 2 a( a 2 b 2 )
  17. 2a3 6a+9 a1 4 a 2 +12a+9 = 2a3 3(2a+3 ) a1 (2a+3 ) 2 = 1 2a+3 [ 2a3 3 a1 2a+3 ] = 1 2a+3 [ (2a+3 ) (2a3 ) 3(a1 ) 3(2a+3 ) ] = 1 2a+3 [ 4 a 2 93a+3 3(2a+3 ) ] = 1 2a+3 [ 4 a 2 3a6 3(2a+3 ) ] = 4 a 2 3a6 3 (2a+3 ) 2
  18. x+1 x 2 +x+1 x1 x 2 x+1 = (x+1 ) ( x 2 x+1 ) (x1 ) ( x 2 +x+1 ) ( x 2 +x+1 ) ( x 2 x+1 ) = x 3 +1( x 3 1 ) ( x 2 +x+1 ) ( x 2 x+1 ) = x 3 +1 x 3 +1 ( x 2 +x+1 ) ( x 2 x+1 ) = 2 ( x 2 +x+1 ) ( x 2 x+1 )
  19. a1 a 2 +a 1 2a2 1 2a+2 = a1 a(a+1 ) 1 2(a1 ) 1 2(a+1 ) = 2 (a1 ) 2 a(a+1 ) a(a1 ) 2a(a+1 ) (a1 ) = 2( a 2 2a+1 ) a 2 a a 2 + a 2a(a+1 ) (a1 ) = 2 a 2 4a+2 2 a 2 2a(a+1 ) (a1 ) = 24a 2a(a+1 ) (a1 ) = 2 (12a ) 2 a(a+1 ) (a1 ) = 12a a( a 2 1 )
  20. 1 4a+4 1 8a8 1 12 a 2 +12 = 1 4(a+1 ) 1 8(a1 ) 1 12( a 2 +1 ) = 1 4 [ 1 a+1 1 2(a1 ) 1 3( a 2 +1 ) ] = 1 4 [ 6(a1 ) ( a 2 +1 ) 3(a+1 ) ( a 2 +1 ) 2(a+1 ) (a1 ) 6(a+1 ) (a1 ) ( a 2 +1 ) ] = 1 4 [ 6( a 3 +a a 2 1 ) 3( a 3 +a+ a 2 +1 ) 2( a 2 1 ) 6(a+1 ) (a1 ) ( a 2 +1 ) ] = 1 4 [ 6 a 3 +6a6 a 2 63 a 3 3a3 a 2 32 a 2 +2 6(a+1 ) (a1 ) ( a 2 +1 ) ] = 1 4 [ 3 a 3 11 a 2 +3a7 6(a+1 ) (a1 ) ( a 2 +1 ) ] = 3 a 3 11 a 2 +3a7 24(a+1 ) (a1 ) ( a 2 +1 )
  21. y x 2 xy 1 x 1 xy = y x(xy ) 1 x 1 xy = y(xy ) x x(xy ) = yx+yx x(xy ) = 2x+2y x(xy ) = 2 (xy ) x (xy ) = 2 x
  22. a a 2 +ab 1 a 1 a+b = a a(a+b ) 1 a 1 a+b = a (a+b ) a a(a+b ) = (a+b ) a (a+b ) = 1 a
  23. 1 x 2 xy 1 x 2 +xy 2y x 3 x y 2 = 1 x(xy ) 1 x(x+y ) 2y x( x 2 y 2 ) = x+y(xy ) 2y x(xy ) (x+y ) = x + y x + y 2y x(xy ) (x+y ) =0
  24. x x 2 +x2 3 x 2 +2x3 x x 2 +5x+6 = x (x+2 ) (x1 ) 3 (x+3 ) (x1 ) x (x+3 ) (x+2 ) = x(x+3 ) 3(x+2 ) x(x1 ) (x+2 ) (x1 ) (x+3 ) = x 2 + 3x 3x 6 x 2 +x (x+2 ) (x1 ) (x+3 ) = x6 (x+2 ) (x1 ) (x+3 )
  25. 3 x 2 +x+1 x+2 (x1 ) 2 19x ( x 3 1 ) (x1 ) = 3 x 2 +x+1 x+2 (x1 ) 2 19x (x1 ) ( x 2 +x+1 ) (x1 ) = 3 x 2 +x+1 x+2 (x1 ) 2 19x (x1 ) 2 ( x 2 +x+1 ) = 3 (x1 ) 2 (x+2 ) ( x 2 +x+1 ) (19x ) (x1 ) 2 ( x 2 +x+1 ) = 3( x 2 2x+1 ) ( x 3 + x 2 +x+2 x 2 +2x+2 ) 1+9x (x1 ) 2 ( x 2 +x+1 ) = 3 x 2 6x + 3 x 3 3 x 2 3x 2 1 + 9x (x1 ) 2 ( x 2 +x+1 ) = x 3 (x1 ) ( x 3 1 )
  26. a 2 + b 2 a 3 b 3 a+b 2 a 2 +2ab+2 b 2 1 2a2b = a 2 + b 2 (ab ) ( a 2 +ab+ b 2 ) a+b 2( a 2 +ab+ b 2 ) 1 2(ab ) = 2( a 2 + b 2 ) (a+b ) (ab ) ( a 2 +ab+ b 2 ) 2(ab ) ( a 2 +ab+ b 2 ) = 2 a 2 +2 b 2 ( a 2 b 2 ) a 2 ab b 2 2(ab ) ( a 2 +ab+ b 2 ) = a 2 + b 2 a 2 + b 2 ab 2(ab ) ( a 2 +ab+ b 2 ) = 2 b 2 ab 2( a 3 b 3 ) = b(2ba ) 2( a 3 b 3 )
  27. 3a 2 a 2 2a4 a1 4 a 2 +8a32 10a1 8 a 2 +40a+32 = 3a 2( a 2 a2 ) a1 4( a 2 +2a8 ) 10a1 8( a 2 +5a+4 ) = 1 2 [ 3a (a2 ) (a+1 ) a1 2(a+4 ) (a2 ) 10a1 4(a+4 ) (a+1 ) ] = 1 2 [ 12a(a+4 ) 2(a1 ) (a+1 ) (10a1 ) (a2 ) 4(a2 ) (a+1 ) (a+4 ) ] = 1 2 [ 12 a 2 +48a2( a 2 1 ) (10 a 2 20aa+2 ) 4(a2 ) (a+1 ) (a+4 ) ] = 1 2 [ 12 a 2 +48a 2 a 2 + 2 10 a 2 +21a 2 4(a2 ) (a+1 ) (a+4 ) ] = 1 2 [ 69a 4(a2 ) (a+1 ) (a+4 ) ] = 69a 8(a2 ) (a+1 ) (a+4 )
  28. 1 4a12x a 2 +9 x 2 a 3 27 x 3 a 2( a 2 +3ax+9 x 2 ) = 1 4(a3x ) a 2 +9 x 2 (a3x ) ( a 2 +3ax+9 x 2 ) a 2( a 2 +3ax+9 x 2 ) = a 2 +3ax+9 x 2 4( a 2 +9 x 2 ) 2a(a3x ) 4(a3x ) ( a 2 +3ax+9 x 2 ) = a 2 +3ax+9 x 2 4 a 2 36 x 2 2 a 2 +6ax 4(a3x ) ( a 2 +3ax+9 x 2 ) = 5 a 2 +9ax27 x 2 4(a3x ) ( a 2 +3ax+9 x 2 ) = 5 a 2 9ax+27 x 2 4( a 3 27 x 3 )
  29. 2 a 2 3 10a+10 a+1 50 9 a 2 14 50a+50 = 2 a 2 3 10(a+1 ) a+1 50 9 a 2 14 50(a+1 ) = 1 10 [ 2 a 2 3 a+1 a+1 5 9 a 2 14 5(a+1 ) ] = 1 10 [ 5(2 a 2 3 ) (a+1 ) 2 (9 a 2 14 ) 5(a+1 ) ] = 1 10 [ 10 a 2 15( a 2 +2a+1 ) 9 a 2 +14 5(a+1 ) ] = 1 10 [ a 2 1 a 2 2a1 5(a+1 ) ] = 1 10 [ 22a 5(a+1 ) ] = 1 [ 2 (1+a ) 5 (a+1 ) ] = 1 25