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


Solution:
Given

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

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

$= \dfrac{\dfrac{(1-xy)^a}{y^a} × \dfrac{(xy+1)^a}{y^a}} {\dfrac{(xy+1)^a}{x^a} × \dfrac{(1-xy)^a}{x^a}}$

$= \dfrac{\dfrac{(1-xy)^a}{y^a} × \dfrac{(1+xy)^a}{y^a}} {\dfrac{(1+xy)^a}{x^a} × \dfrac{(1-xy)^a}{x^a}}$

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

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

$= \dfrac{x^a.x^a}{y^a.y^a}$

$= \dfrac{x^{a+a}}{y^{a+a}}$

$= \dfrac{x^{2a}}{y^{2a}}$

$= \left ( \dfrac{x}{y} \right )^{2a}$
= Answer

Related Notes and Solutions:

Here is the website link to the notes of Indices.

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