Question: $\sqrt[\frac{1}{ab}]{\dfrac{x^{\frac{1}{a} } } { x^{\frac{1}{b} } } } ×$$\sqrt[\frac{1}{bc}]{\dfrac{x^{\frac{1}{b} } } { x^{\frac{1}{c} } } } ×$$\sqrt[\frac{1}{ca}]{\dfrac{x^{\frac{1}{c} } } { x^{\frac{1}{a} } } }$

Solution:
Given

$= \sqrt[\frac{1}{ab}]{\dfrac{x^{\frac{1}{a} } } { x^{\frac{1}{b} } } } ×\sqrt[\frac{1}{bc}]{\dfrac{x^{\frac{1}{b} } } { x^{\frac{1}{c} } } } ×\sqrt[\frac{1}{ca}]{\dfrac{x^{\frac{1}{c} } } { x^{\frac{1}{a} } } }$

$= \sqrt[\frac{1}{ab}]{ x^{ \frac{1}{a} -\frac{1}{b} } } × \sqrt[\frac{1}{bc}]{ x^{ \frac{1}{b} -\frac{1}{c} } } × \sqrt[\frac{1}{ca}]{ x^{ \frac{1}{c} -\frac{1}{a} } }$

$= \sqrt[\frac{1}{ab}]{ x^{ \frac{b-a}{ab} } } × \sqrt[\frac{1}{bc}]{ x^{ \frac{c-b}{bc} } } × \sqrt[\frac{1}{ca}]{ x^{ \frac{a-c}{ca} } }$

$= \left ( x^{ \dfrac{\frac{b-a}{ab}} {\frac{1}{ab}} } \right ) × \left ( x^{ \dfrac{\frac{c-b}{bc}} {\frac{1}{bc}} } \right ) ×  \left ( x^{ \dfrac{\frac{a-c}{ca}} {\frac{1}{ca}} } \right )$

$= x^{b-a} × x^{c-b} × x^{a-c}$

$= x^{(b-a)+(c-b) +(a-c)}$

$= x^{b-a+c-b+a-c}$

$= x^0$

$= 1$
= Answer


Related Notes and Solutions:

Here is the website link to the notes of Indices.

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