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Simplificación de expresiones/ Descomposición en fracciones simples/ (y/(x*y-x^2)+x/(x*y-y^2))/(((x^2+2*x*y+y^2)/(1/x+1/y)))

¿Cómo vas a descomponer esta (y/(x*y-x^2)+x/(x*y-y^2))/(((x^2+2*x*y+y^2)/(1/x+1/y))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   y          x    
-------- + --------
       2          2
x*y - x    x*y - y 
-------------------
 / 2            2\ 
 |x  + 2*x*y + y | 
 |---------------| 
 |     1   1     | 
 |     - + -     | 
 \     x   y     /
xxy−y2+y−x2+xy11y+1x(y2+(x2+2xy))\frac{\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}}{\frac{1}{\frac{1}{y} + \frac{1}{x}} \left(y^{2} + \left(x^{2} + 2 x y\right)\right)} 
(y/(x*y - x^2) + x/(x*y - y^2))/(((x^2 + (2*x)*y + y^2)/(1/x + 1/y)))
Simplificación general [src] 
  1  
-----
 2  2
x *y
1x2y2\frac{1}{x^{2} y^{2}} 
1/(x^2*y^2)
Respuesta numérica [src] 
(1/x + 1/y)*(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2 + 2.0*x*y)
(1/x + 1/y)*(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2 + 2.0*x*y)
Denominador racional [src] 
           /  /   2      \     /   2      \\   
   (x + y)*\x*\- x  + x*y/ + y*\- y  + x*y//   
-----------------------------------------------
    /   2      \ /   2      \ / 2    2        \
x*y*\- x  + x*y/*\- y  + x*y/*\x  + y  + 2*x*y/
(x+y)(x(−x2+xy)+y(xy−y2))xy(−x2+xy)(xy−y2)(x2+2xy+y2)\frac{\left(x + y\right) \left(x \left(- x^{2} + x y\right) + y \left(x y - y^{2}\right)\right)}{x y \left(- x^{2} + x y\right) \left(x y - y^{2}\right) \left(x^{2} + 2 x y + y^{2}\right)} 
(x + y)*(x*(-x^2 + x*y) + y*(-y^2 + x*y))/(x*y*(-x^2 + x*y)*(-y^2 + x*y)*(x^2 + y^2 + 2*x*y))
Abrimos la expresión [src] 
/1   1\ /   y          x    \
|- + -|*|-------- + --------|
\x   y/ |       2          2|
        \x*y - x    x*y - y /
-----------------------------
        2            2       
       x  + 2*x*y + y
(1y+1x)(xxy−y2+y−x2+xy)y2+(x2+2xy)\frac{\left(\frac{1}{y} + \frac{1}{x}\right) \left(\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}\right)}{y^{2} + \left(x^{2} + 2 x y\right)} 
(1/x + 1/y)*(y/(x*y - x^2) + x/(x*y - y^2))/(x^2 + (2*x)*y + y^2)
Denominador común [src] 
  1  
-----
 2  2
x *y
1x2y2\frac{1}{x^{2} y^{2}} 
1/(x^2*y^2)
Potencias [src] 
/1   1\ /    x            y     \
|- + -|*|---------- + ----------|
\x   y/ |   2            2      |
        \- y  + x*y   - x  + x*y/
---------------------------------
          2    2                 
         x  + y  + 2*x*y
(1y+1x)(xxy−y2+y−x2+xy)x2+2xy+y2\frac{\left(\frac{1}{y} + \frac{1}{x}\right) \left(\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}\right)}{x^{2} + 2 x y + y^{2}} 
(1/x + 1/y)*(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2 + 2*x*y)
Parte trigonométrica [src] 
/1   1\ /    x            y     \
|- + -|*|---------- + ----------|
\x   y/ |   2            2      |
        \- y  + x*y   - x  + x*y/
---------------------------------
          2    2                 
         x  + y  + 2*x*y
(1y+1x)(xxy−y2+y−x2+xy)x2+2xy+y2\frac{\left(\frac{1}{y} + \frac{1}{x}\right) \left(\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}\right)}{x^{2} + 2 x y + y^{2}} 
(1/x + 1/y)*(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2 + 2*x*y)
Combinatoria [src] 
  1  
-----
 2  2
x *y
1x2y2\frac{1}{x^{2} y^{2}} 
1/(x^2*y^2)
Compilar la expresión [src] 
/1   1\ /    x            y     \
|- + -|*|---------- + ----------|
\x   y/ |   2            2      |
        \- y  + x*y   - x  + x*y/
---------------------------------
          2    2                 
         x  + y  + 2*x*y
(1y+1x)(xxy−y2+y−x2+xy)x2+2xy+y2\frac{\left(\frac{1}{y} + \frac{1}{x}\right) \left(\frac{x}{x y - y^{2}} + \frac{y}{- x^{2} + x y}\right)}{x^{2} + 2 x y + y^{2}} 
(1/x + 1/y)*(x/(-y^2 + x*y) + y/(-x^2 + x*y))/(x^2 + y^2 + 2*x*y)
Unión de expresiones racionales [src] 
           / 2            2        \    
   (x + y)*\x *(y - x) + y *(x - y)/    
----------------------------------------
 2  2                 / 2              \
x *y *(x - y)*(y - x)*\y  + x*(x + 2*y)/
(x+y)(x2(−x+y)+y2(x−y))x2y2(−x+y)(x−y)(x(x+2y)+y2)\frac{\left(x + y\right) \left(x^{2} \left(- x + y\right) + y^{2} \left(x - y\right)\right)}{x^{2} y^{2} \left(- x + y\right) \left(x - y\right) \left(x \left(x + 2 y\right) + y^{2}\right)} 
(x + y)*(x^2*(y - x) + y^2*(x - y))/(x^2*y^2*(x - y)*(y - x)*(y^2 + x*(x + 2*y)))
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