Ejercicio 131

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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