Question: Simplify: $\sqrt[xy]{\dfrac{a^x}{a^y}} ×$$\sqrt[yz]{\dfrac{a^y}{a^z}} ×$$\sqrt[zx]{\dfrac{a^z}{a^x}} $

Solution:
Given,

$= \sqrt[xy]{\dfrac{a^x}{a^y}} × \sqrt[yz]{\dfrac{a^y}{a^z}} × \sqrt[zx]{\dfrac{a^z}{a^x}}$

$= \left ( x^{\dfrac{x-y}{xy}} \right ) × \left ( x^{\dfrac{y-z}{yz}} \right ) × \left ( x^{\dfrac{z-x}{zx}} \right ) $

$= x^{\dfrac{x-y}{xy} + \dfrac{y-z}{yz} + \dfrac{z-x}{zx}}$

$= x^{\dfrac{(x-y)z + x(y-z)}{xyz} +\dfrac{z-x}{zx}}$

$= x^{\dfrac{xz -yz +xy -xz}{xyz} +\dfrac{z-x}{zx}}$

$= x^{\dfrac{xy -yz}{xyz} +\dfrac{z-x}{zx}}$

$= x^{\dfrac{xy -yz + y(z-x)}{xyz}}$

$= x^{\dfrac{xy -yz +yz -xy}{xyz}}$

$= x^{\dfrac{0}{xyz}}$

$= x^0$

$= 1$
= Answer

Related Notes and Solutions:

Here is the website link to the notes of Indices.

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