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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