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CAPITULO XIV

Operaciones con Fracciones
Ejercicio 127
Simplificar:
  1. 1 a+1 + 1 a1 = a 1 +a+ 1 (a+1 ) (a1 ) = 2a a 2 1
  2. 2 x+4 + 1 x3 = 2(x3 ) +x+4 (x+4 ) (x3 ) = 2x6+x+4 (x+4 ) (x3 ) = 3x2 (x+4 ) (x3 )
  3. 3 1x + 6 2x+5 = 3(2x+5 ) +6(1x ) (1x ) (2x+5 ) = 6x +15+6 6x (1x ) (2x+5 ) = 21 (1x ) (2x+5 )
  4. x xy + x x+y = x(x+y ) +x(xy ) (xy ) (x+y ) = x[x+ y +x y ] x 2 y 2 = 2 x 2 x 2 y 2
  5. m+3 m3 + m+2 m2 = (m2 ) (m+3 ) +(m+2 ) (m3 ) (m3 ) (m2 ) = m 2 + m 6+ m 2 m 6 (m3 ) (m2 ) = 2 m 2 12 (m3 ) (m2 ) = 2( m 2 6 ) (m3 ) (m2 )
  6. x+y xy + xy x+y = (x+y ) 2 + (xy ) 2 (xy ) (x+y ) = x 2 + 2xy + y 2 + x 2 2xy + y 2 x 2 y 2 = 2 x 2 +2 y 2 x 2 y 2 = 2( x 2 + y 2 ) x 2 y 2
  7. x x 2 1 + x+1 (x1 ) 2 = x (x1 ) (x+1 ) + x+1 (x1 ) 2 = x(x1 ) + (x+1 ) 2 (x+1 ) (x1 ) 2 = x 2 x+ x 2 +2x+1 (x+1 ) (x1 ) 2 = 2 x 2 +x+1 (x+1 ) (x1 ) 2
  8. 2 x5 + 3x x 2 25 = 2 x5 + 3x (x5 ) (x+5 ) = 2(x+5 ) +3x (x5 ) (x+5 ) = 2x+10+3x (x5 ) (x+5 ) = 5x+10 (x5 ) (x+5 ) = 5(x+2 ) (x5 ) (x+5 )
  9. 1 3x2y + xy 9 x 2 4 y 2 = 1 3x2y + xy (3x2y ) (3x+2y ) = 3x+2y+xy (3x2y ) (3x+2y ) = 4x+y (3x2y ) (3x+2y )
  10. x+a x+3a + 3 a 2 x 2 x 2 9 a 2 = x+a x+3a + 3 a 2 x 2 (x+3a ) (x3a ) = (x+a ) (x3a ) +3 a 2 x 2 (x+3a ) (x3a ) = x 2 2ax 3 a 2 + 3 a 2 x 2 (x+3a ) (x3a ) = 2ax x 2 9 a 2 = 2ax 9 a 2 x 2
  11. a 1 a 2 + a 1+ a 2 =a[ 1+ a 2 +1 a 2 (1 a 2 ) (1+ a 2 ) ] =a[ 2 1 a 4 ] = 2a 1 a 4
  12. 2 a 2 ab + 2 ab+ b 2 =2[ 1 a(ab ) + 1 b(a+b ) ] =2[ b(a+b ) +a(ab ) ab(a+b ) (ab ) ] =2[ ab + b 2 + a 2 ab ab( a 2 b 2 ) ] = 2( a 2 + b 2 ) ab( a 2 b 2 )
  13. ab 9 a 2 b 2 + a 3a+b =a[ b (3a+b ) (3ab ) + 1 3a+b ] =a[ b +3a b (3a+b ) (3ab ) ] = 3 a 2 9 a 2 b 2
  14. 1 a 2 b 2 + 1 (ab ) 2 = 1 (a+b ) (ab ) + 1 (ab ) 2 = a b +a+ b (a+b ) (ab ) 2 = 2a (a+b ) (ab ) 2
  15. 3 x 2 + y 2 + 2 (x+y ) 2 = 3 (x+y ) 2 +2( x 2 + y 2 ) ( x 2 + y 2 ) (x+y ) 2 = 3( x 2 +2xy+ y 2 ) +2 x 2 +2 y 2 ( x 2 + y 2 ) (x+y ) 2 = 3 x 2 +6xy+3 y 2 +2 x 2 +2 y 2 ( x 2 + y 2 ) (x+y ) 2 = 5 x 2 +6xy+5 y 2 ( x 2 + y 2 ) (x+y ) 2
  16. x a 2 ax + a+x ax + a ax x 2 = x a(ax ) + a+x ax + a x(ax ) = x 2 +(a+x ) (ax ) + a 2 ax(ax ) = x 2 + a 2 x 2 + a 2 ax(ax ) = 2 a 2 a x(ax ) = 2a x(ax )
  17. 3 2x+4 + x1 2x4 + x+8 x 2 4 = 3 2(x+2 ) + x1 2(x2 ) + x+8 (x+2 ) (x2 ) = 3(x2 ) +(x1 ) (x+2 ) +2(x+8 ) 2(x+2 ) (x2 ) = 3x6+ x 2 +x2+2x+16 2(x+2 ) (x2 ) = x 2 +6x+8 2(x+2 ) (x2 ) = (x+4 )(x+2 ) 2 (x+2 )(x2 ) = x+4 2(x2 )
  18. 1 x+ x 2 + 1 x x 2 + x+3 1 x 2 = 1 x(1+x ) + 1 x(1x ) + x+3 (1x ) (1+x ) = 1 x +1+ x +x(x+3 ) x(1x ) (1+x ) = 2+ x 2 +3x x(1x ) (1+x ) = x 2 +3x+2 x(1x ) (1+x ) = (x+2 )(x+1 ) x(1x )(x+1 ) = x+2 x(1x )
  19. xy x+y + x+y xy + 4xy x 2 y 2 = xy x+y + x+y xy + 4xy (x+y ) (xy ) = (xy ) 2 + (x+y ) 2 +4xy (x+y ) (xy ) = x 2 2xy + y 2 + x 2 + 2xy + y 2 +4xy (x+y ) (xy ) = 2 x 2 +4xy+2 y 2 (x+y ) (xy ) = 2( x 2 +2xy+ y 2 ) (x+y ) (xy ) = 2 (x+y ) 2 (x+y )(xy ) = 2(x+y ) xy
  20. 1 a5 + a a 2 4a5 + a+5 a 2 +2a+1 = 1 a5 + a (a5 ) (a+1 ) + a+5 (a+1 ) 2 = (a+1 ) 2 +a(a+1 ) +(a+5 ) (a5 ) (a5 ) (a+1 ) 2 = a 2 +2a+1+ a 2 +a+ a 2 25 (a5 ) (a+1 ) 2 = 3 a 2 +3a24 (a5 ) (a+1 ) 2 = 3( a 2 +a8 ) (a5 ) (a+1 ) 2
  21. 3 a + 2 5a3 + 185a 25 a 2 9 = 3 a + 2 5a3 + 185a (5a3 ) (5a+3 ) = 3(5a3 ) (5a+3 ) +2a(5a+3 ) +a(185a ) a(5a3 ) (5a+3 ) = 3(25 a 2 9 ) +10 a 2 +6a+a85 a 2 a(5a3 ) (5a+3 ) = 75 a 2 27+7a 75 a 2 a(25 a 2 9 ) = 7a27 a(25 a 2 9 )
  22. x+1 10 + x3 5x10 + x2 2 = x+1 10 + x3 5(x2 ) + x2 2 = (x+1 ) (x2 ) +2(x3 ) +5 (x2 ) 2 10(x2 ) = x 2 x2+2x6+5( x 2 4x+4 ) 10(x2 ) = x 2 +x8+5 x 2 20x+20 10(x2 ) = 6 x 2 19x+12 10(x2 )
  23. x+5 x 2 +x12 + x+4 x 2 +2x15 + x3 x 2 +9x+20 = x+5 (x+4 ) (x3 ) + x+4 (x+5 ) (x3 ) + x3 (x+5 ) (x+4 ) = (x+5 ) 2 + (x+4 ) 2 + (x3 ) 2 (x+4 ) (x3 ) (x+5 ) = x 2 +10x+25+ x 2 +8x+16+ x 2 6x+9 (x+4 ) (x3 ) (x+5 ) = 3 x 2 +12x+50 (x+4 ) (x3 ) (x+5 )
  24. 1 x2 + 12 x 2 x 3 8 + x x 2 +2x+4 = 1 x2 + 12 x 2 (x2 ) ( x 2 +2x+4 ) + x x 2 +2x+4 = x 2 +2x+4+12 x 2 +x(x2 ) (x2 ) ( x 2 +2x+4 ) = x 2 + 2x +4+1 2 x 2 + x 2 2x x 3 8 = 5 x 3 8
  25. 2 a+1 + a (a+1 ) 2 + a+1 (a+1 ) 3 = 1 a+1 [2+ a a+1 + a+1 (a+1 ) 2 ] = 1 a+1 [ 2 (a+1 ) 2 +a(a+1 ) +a+1 (a+1 ) 2 ] = 1 a+1 [ 2( a 2 +2a+1 ) + a 2 +a+a+1 (a+1 ) 2 ] = 1 a+1 [ 2 a 2 +4a+2+ a 2 +2a+1 (a+1 ) 2 ] = 1 a+1 [ 3 a 2 +6a+3 (a+1 ) 2 ] = 3 a+1 [ a 2 +2a+1 (a+1 ) 2 ] = 3 a+1 [ (a+1 ) 2 (a+1 ) 2 ] = 3 a+1
  26. 2x 3 x 2 +11x+6 + x+1 x 2 9 + 1 3x+2 = 2x 3 x 2 +9x+2x+6 + x+1 (x3 ) (x+3 ) + 1 3x+2 = 2x 3x(x+3 ) +2(x+3 ) + x+1 (x3 ) (x+3 ) + 1 3x+2 = 2x (3x+2 ) (x+3 ) + x+1 (x3 ) (x+3 ) + 1 3x+2 = 2x(x3 ) +(x+1 ) (3x+2 ) +(x+3 ) (x3 ) (3x+2 ) (x+3 ) (x3 ) = 6 x 2 x7 (3x+2 ) (x+3 ) (x3 )
  27. x 2 4 x 3 +1 + 1 x+1 + 3 x 2 x+1 = x 2 4 (x+1 ) ( x 2 x+1 ) + 1 x+1 + 3 x 2 x+1 = x 2 4+ x 2 x+1+3(x+1 ) (x+1 ) ( x 2 x+1 ) = 2 x 2 4 x+ 1 +3x+ 3 (x+1 ) ( x 2 x+1 ) = 2 x 2 +2x (x+1 ) ( x 2 x+1 ) = 2x (x+1 ) (x+1 )( x 2 x+1 ) = 2x x 2 x+1
  28. 1 x1 + 1 (x1 ) (x+2 ) + x+1 (x1 ) (x+2 ) (x+3 ) = 1 x1 [1+ 1 x+2 + x+1 (x+2 ) (x+3 ) ] = 1 x1 [ (x+2 ) (x+3 ) +x+3+x+1 (x+2 ) (x+3 ) ] = 1 x1 [ x 2 +5x+6+2x+4 (x+2 ) (x+3 ) ] = 1 x1 [ x 2 +7x+10 (x+2 ) (x+3 ) ] = 1 x1 [ (x+2 )(x+5 ) (x+2 )(x+3 ) ] = x+5 (x1 ) (x+3 )
  29. x2 2 x 2 5x3 + x3 2 x 2 3x2 + 2x1 x 2 5x+6 = x2 2 x 2 +x6x3 + x3 2 x 2 4x+x2 + 2x1 (x3 ) (x2 ) = x2 x(2x+1 ) 3(2x+1 ) + x3 2x(x2 ) +(x2 ) + 2x1 (x3 ) (x2 ) = x2 (x3 ) (2x+1 ) + x3 (2x+1 ) (x2 ) + 2x1 (x3 ) (x2 ) = (x2 ) 2 + (x3 ) 2 +(2x+1 ) (2x1 ) (x3 ) (2x+1 ) (x2 ) = x 2 4x+4+ x 2 6x+9+4 x 2 1 (x3 ) (2x+1 ) (x2 ) = 6 x 2 10x+12 (x3 ) (2x+1 ) (x2 ) = 2(3 x 2 5x+6 ) (x3 ) (2x+1 ) (x2 )
  30. a2 a1 + a+3 a+2 + a+1 a3 = (a2 ) (a+2 ) (a3 ) +(a1 ) (a3 ) (a+3 ) +(a1 ) (a+1 ) (a+2 ) (a1 ) (a+2 ) (a3 ) = ( a 2 4 ) (a3 ) +(a1 ) ( a 2 9 ) +( a 2 1 ) (a+2 ) (a1 ) (a+2 ) (a3 ) = a 3 4a3 a 2 +12+ a 3 9a a 2 +9+ a 3 +2 a 2 a2 (a1 ) (a+2 ) (a3 ) = 3 a 3 2 a 2 14a+19 (a1 ) (a+2 ) (a3 )