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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