Question: Simplify: $\dfrac{ \left ( 1 + \dfrac{x}{y} \right ) ^ {\frac{x}{x-y}} × \left ( 1 - \dfrac{y}{x} \right ) ^{\frac{y}{x-y}} }  {  \left ( \dfrac{y}{x} +1 \right ) ^ {\frac{x}{x-y}} × \left ( \dfrac{x}{y} -1 \right ) ^{\frac{y}{x-y}}  }$


Solution:
Given,

$= \dfrac{ \left ( 1 + \dfrac{x}{y} \right ) ^ {\frac{x}{x-y}} × \left ( 1 - \dfrac{y}{x} \right ) ^{\frac{y}{x-y}} }  {  \left ( \dfrac{y}{x} +1 \right ) ^ {\frac{x}{x-y}} × \left ( \dfrac{x}{y} -1 \right ) ^{\frac{y}{x-y}}  }$

$= \dfrac{ \left ( \dfrac{y+x}{y} \right ) ^ {\frac{x}{x-y}} × \left ( \dfrac{x-y}{x} \right ) ^{\frac{y}{x-y}} }  {  \left ( \dfrac{y+x}{x} +\right ) ^ {\frac{x}{x-y}} × \left ( \dfrac{x-y}{y} \right ) ^{\frac{y}{x-y}}  }$

$= \dfrac{\dfrac{(y+x)^{\frac{x}{x-y}}}{(y)^{\frac{x}{x-y}} }   × \dfrac{(x-y)^{ \frac{y}{x-y}} }{(x)^{\frac{y}{x-y}} }  }  {  \dfrac{(y+x)^{\frac{x}{x-y} } }{(x)^{ \frac{x}{x-y}}}  × \dfrac{(x-y)^{\frac{y}{x-y} }}{(y)^{ \frac{y}{x-y}}} }$

$= \dfrac{ \dfrac{1} { (y)^{\frac{x}{x-y} } }  × \dfrac{1}{ (x)^{\frac{y}{x-y} }}}  { \dfrac{1} { (x)^{\frac{x}{x-y} } }  × \dfrac{1}{ (y)^{\frac{y}{x-y} }} }$

$= \dfrac{x^{\frac{x}{x-y}} × y^{\frac{y}{x-y}}}{y^{\frac{x}{x-y}} × x^{\frac{y}{x-y}}}$

$= \dfrac{x^{\frac{x}{x-y}} × x^{- \frac{y}{x-y}}}{y^{\frac{x}{x-y}} × y^{- \frac{y}{x-y}}}$

$= \dfrac{x^{\frac{x}{x-y} - \frac{y}{x-y}}}{y^{\frac{x}{x-y} - \frac{y}{x-y}}}$

$= \dfrac{x^{\frac{x-y}{x-y}}}{y^{\frac{x-y}{x-y}}}$

$= \dfrac{x^1}{y^1}$

$= \dfrac{x}{y}$
= Answer

Related Notes and Solutions:

Here is the website link to the notes of Indices.

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