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Simplificación de expresiones/ Descomposición en fracciones simples/ (y+z+y*z/x)*(x+y+x*y/z+x+z+x*z/y)/(2*(x+y+z)+y*z/x+x*y/z+x*z/y)

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

Expresión a simplificar:

Solución

Ha introducido [src] 
/        y*z\ /        x*y           x*z\
|y + z + ---|*|x + y + --- + x + z + ---|
\         x / \         z             y /
-----------------------------------------
                     y*z   x*y   x*z     
     2*(x + y + z) + --- + --- + ---     
                      x     z     y
((y+z)+yzx)((z+(x+((x+y)+xyz)))+xzy)((2(z+(x+y))+yzx)+xyz)+xzy\frac{\left(\left(y + z\right) + \frac{y z}{x}\right) \left(\left(z + \left(x + \left(\left(x + y\right) + \frac{x y}{z}\right)\right)\right) + \frac{x z}{y}\right)}{\left(\left(2 \left(z + \left(x + y\right)\right) + \frac{y z}{x}\right) + \frac{x y}{z}\right) + \frac{x z}{y}} 
((y + z + (y*z)/x)*(x + y + (x*y)/z + x + z + (x*z)/y))/(2*(x + y + z) + (y*z)/x + (x*y)/z + (x*z)/y)
Simplificación general [src] 
y + z
y+zy + z 
y + z
Respuesta numérica [src] 
(y + z + y*z/x)*(y + z + 2*x + x*y/z + x*z/y)/(2.0*x + 2.0*y + 2.0*z + x*y/z + x*z/y + y*z/x)
(y + z + y*z/x)*(y + z + 2*x + x*y/z + x*z/y)/(2.0*x + 2.0*y + 2.0*z + x*y/z + x*z/y + y*z/x)
Potencias [src] 
/        y*z\ /              x*y   x*z\
|y + z + ---|*|y + z + 2*x + --- + ---|
\         x / \               z     y /
---------------------------------------
                     x*y   x*z   y*z   
   2*x + 2*y + 2*z + --- + --- + ---   
                      z     y     x
(y+z+yzx)(xyz+2x+xzy+y+z)xyz+2x+xzy+2y+2z+yzx\frac{\left(y + z + \frac{y z}{x}\right) \left(\frac{x y}{z} + 2 x + \frac{x z}{y} + y + z\right)}{\frac{x y}{z} + 2 x + \frac{x z}{y} + 2 y + 2 z + \frac{y z}{x}} 
(y + z + y*z/x)*(y + z + 2*x + x*y/z + x*z/y)/(2*x + 2*y + 2*z + x*y/z + x*z/y + y*z/x)
Combinatoria [src] 
y + z
y+zy + z 
y + z
Denominador racional [src] 
 2  3    2  3    2  3    3  2          3          3        2  2        2  2        2  2
x *y  + x *z  + y *z  + y *z  + 2*x*y*z  + 2*x*z*y  + 3*y*x *z  + 3*z*x *y  + 4*x*y *z 
---------------------------------------------------------------------------------------
                  2  2    2  2    2  2          2          2          2                
                 x *y  + x *z  + y *z  + 2*x*y*z  + 2*x*z*y  + 2*y*z*x
x2y3+3x2y2z+3x2yz2+x2z3+2xy3z+4xy2z2+2xyz3+y3z2+y2z3x2y2+2x2yz+x2z2+2xy2z+2xyz2+y2z2\frac{x^{2} y^{3} + 3 x^{2} y^{2} z + 3 x^{2} y z^{2} + x^{2} z^{3} + 2 x y^{3} z + 4 x y^{2} z^{2} + 2 x y z^{3} + y^{3} z^{2} + y^{2} z^{3}}{x^{2} y^{2} + 2 x^{2} y z + x^{2} z^{2} + 2 x y^{2} z + 2 x y z^{2} + y^{2} z^{2}} 
(x^2*y^3 + x^2*z^3 + y^2*z^3 + y^3*z^2 + 2*x*y*z^3 + 2*x*z*y^3 + 3*y*x^2*z^2 + 3*z*x^2*y^2 + 4*x*y^2*z^2)/(x^2*y^2 + x^2*z^2 + y^2*z^2 + 2*x*y*z^2 + 2*x*z*y^2 + 2*y*z*x^2)
Compilar la expresión [src] 
/      /    z\\ /            /    x\   x*z\
|z + y*|1 + -||*|z + 2*x + y*|1 + -| + ---|
\      \    x// \            \    z/    y /
-------------------------------------------
                    /    x   z\   x*z      
      2*x + 2*z + y*|2 + - + -| + ---      
                    \    z   x/    y
(y(1+zx)+z)(2x+xzy+y(xz+1)+z)2x+xzy+y(xz+2+zx)+2z\frac{\left(y \left(1 + \frac{z}{x}\right) + z\right) \left(2 x + \frac{x z}{y} + y \left(\frac{x}{z} + 1\right) + z\right)}{2 x + \frac{x z}{y} + y \left(\frac{x}{z} + 2 + \frac{z}{x}\right) + 2 z} 
/      /    y\\ /      /    y\     /    x\\
|y + z*|1 + -||*|y + x*|2 + -| + z*|1 + -||
\      \    x// \      \    z/     \    y//
-------------------------------------------
              /    y\     /    x   y\      
      2*y + x*|2 + -| + z*|2 + - + -|      
              \    z/     \    y   x/
(y+z(1+yx))(x(yz+2)+y+z(xy+1))x(yz+2)+2y+z(xy+2+yx)\frac{\left(y + z \left(1 + \frac{y}{x}\right)\right) \left(x \left(\frac{y}{z} + 2\right) + y + z \left(\frac{x}{y} + 1\right)\right)}{x \left(\frac{y}{z} + 2\right) + 2 y + z \left(\frac{x}{y} + 2 + \frac{y}{x}\right)} 
/      /    y\\ /            /    x\   x*y\
|y + z*|1 + -||*|y + 2*x + z*|1 + -| + ---|
\      \    x// \            \    y/    z /
-------------------------------------------
                    /    x   y\   x*y      
      2*x + 2*y + z*|2 + - + -| + ---      
                    \    y   x/    z
(y+z(1+yx))(xyz+2x+y+z(xy+1))xyz+2x+2y+z(xy+2+yx)\frac{\left(y + z \left(1 + \frac{y}{x}\right)\right) \left(\frac{x y}{z} + 2 x + y + z \left(\frac{x}{y} + 1\right)\right)}{\frac{x y}{z} + 2 x + 2 y + z \left(\frac{x}{y} + 2 + \frac{y}{x}\right)} 
/      /    z\\ /      /    z\     /    x\\
|z + y*|1 + -||*|z + x*|2 + -| + y*|1 + -||
\      \    x// \      \    y/     \    z//
-------------------------------------------
              /    z\     /    x   z\      
      2*z + x*|2 + -| + y*|2 + - + -|      
              \    y/     \    z   x/
(y(1+zx)+z)(x(2+zy)+y(xz+1)+z)x(2+zy)+y(xz+2+zx)+2z\frac{\left(y \left(1 + \frac{z}{x}\right) + z\right) \left(x \left(2 + \frac{z}{y}\right) + y \left(\frac{x}{z} + 1\right) + z\right)}{x \left(2 + \frac{z}{y}\right) + y \left(\frac{x}{z} + 2 + \frac{z}{x}\right) + 2 z} 
/      /    z\\ /        /    x\     /    x\\
|z + y*|1 + -||*|2*x + y*|1 + -| + z*|1 + -||
\      \    x// \        \    z/     \    y//
---------------------------------------------
               /    x   z\     /    x\       
       2*x + y*|2 + - + -| + z*|2 + -|       
               \    z   x/     \    y/
(y(1+zx)+z)(2x+y(xz+1)+z(xy+1))2x+y(xz+2+zx)+z(xy+2)\frac{\left(y \left(1 + \frac{z}{x}\right) + z\right) \left(2 x + y \left(\frac{x}{z} + 1\right) + z \left(\frac{x}{y} + 1\right)\right)}{2 x + y \left(\frac{x}{z} + 2 + \frac{z}{x}\right) + z \left(\frac{x}{y} + 2\right)} 
/        y*z\ /              x*y   x*z\
|y + z + ---|*|y + z + 2*x + --- + ---|
\         x / \               z     y /
---------------------------------------
                     x*y   x*z   y*z   
   2*x + 2*y + 2*z + --- + --- + ---   
                      z     y     x
(y+z+yzx)(xyz+2x+xzy+y+z)xyz+2x+xzy+2y+2z+yzx\frac{\left(y + z + \frac{y z}{x}\right) \left(\frac{x y}{z} + 2 x + \frac{x z}{y} + y + z\right)}{\frac{x y}{z} + 2 x + \frac{x z}{y} + 2 y + 2 z + \frac{y z}{x}} 
/          /    y   z\\ /        y*z\
|y + z + x*|2 + - + -||*|y + z + ---|
\          \    z   y// \         x /
-------------------------------------
                 /    y   z\   y*z   
   2*y + 2*z + x*|2 + - + -| + ---   
                 \    z   y/    x
(y+z+yzx)(x(yz+2+zy)+y+z)x(yz+2+zy)+2y+2z+yzx\frac{\left(y + z + \frac{y z}{x}\right) \left(x \left(\frac{y}{z} + 2 + \frac{z}{y}\right) + y + z\right)}{x \left(\frac{y}{z} + 2 + \frac{z}{y}\right) + 2 y + 2 z + \frac{y z}{x}} 
(y + z + x*(2 + y/z + z/y))*(y + z + y*z/x)/(2*y + 2*z + x*(2 + y/z + z/y) + y*z/x)
Denominador común [src] 
y + z
y+zy + z 
y + z
Parte trigonométrica [src] 
/        y*z\ /              x*y   x*z\
|y + z + ---|*|y + z + 2*x + --- + ---|
\         x / \               z     y /
---------------------------------------
                     x*y   x*z   y*z   
   2*x + 2*y + 2*z + --- + --- + ---   
                      z     y     x
(y+z+yzx)(xyz+2x+xzy+y+z)xyz+2x+xzy+2y+2z+yzx\frac{\left(y + z + \frac{y z}{x}\right) \left(\frac{x y}{z} + 2 x + \frac{x z}{y} + y + z\right)}{\frac{x y}{z} + 2 x + \frac{x z}{y} + 2 y + 2 z + \frac{y z}{x}} 
(y + z + y*z/x)*(y + z + 2*x + x*y/z + x*z/y)/(2*x + 2*y + 2*z + x*y/z + x*z/y + y*z/x)
Unión de expresiones racionales [src] 
/   2     / 2                        \\                  
\x*z  + y*\z  + x*y + x*z + z*(x + y)//*(x*(y + z) + y*z)
---------------------------------------------------------
         /   2                            \    2  2      
       y*\y*x  + z*(y*z + 2*x*(x + y + z))/ + x *z
(xz2+y(xy+xz+z2+z(x+y)))(x(y+z)+yz)x2z2+y(x2y+z(2x(x+y+z)+yz))\frac{\left(x z^{2} + y \left(x y + x z + z^{2} + z \left(x + y\right)\right)\right) \left(x \left(y + z\right) + y z\right)}{x^{2} z^{2} + y \left(x^{2} y + z \left(2 x \left(x + y + z\right) + y z\right)\right)} 
(x*z^2 + y*(z^2 + x*y + x*z + z*(x + y)))*(x*(y + z) + y*z)/(y*(y*x^2 + z*(y*z + 2*x*(x + y + z))) + x^2*z^2)
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