Question: Simplify: $\left ( \dfrac{x^a}{x.x^{-b}} \right )^{a-b} ×$$\left ( \dfrac{x^b}{x.x^{-c}} \right )^{b-c} ×$$\left ( \dfrac{x^c}{x.x^{-a}} \right )^{c-a}$

Solution:
Given,

$= \left ( \dfrac{x^a}{x.x^{-b}} \right )^{a-b} ×\left ( \dfrac{x^b}{x.x^{-c}} \right )^{b-c} ×\left ( \dfrac{x^c}{x.x^{-a}} \right )^{c-a}$

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

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

$= \left (\dfrac{x^a × x^b}{x} \right )^{a-b} × \left ( \dfrac{x^b × x^c}{x} \right ) ^{b-c} × \left ( \dfrac{x^c × x^a}{x} \right )^{c-a}$

$= \left (\dfrac{x^{a+b}}{x} \right )^{a-b} × \left ( \dfrac{x^{b+c}}{x} \right ) ^{b-c} × \left ( \dfrac{x^{c+a}}{x} \right )^{c-a}$

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

$= \dfrac{x^{a²-b²}}{x^{a-b}}  × \dfrac{x^{b²-c²}}{x^{b-c}} × \dfrac{x^{c²-a²}}{x^{c-a}} $

$= \dfrac{x^{a²-b²} × x^{b²-c²} × x^{c²-a²}}{x^{a-b} × x^{b-c} × x^{c-a}}$

$= \dfrac{x^{a²-b²+b²-c²+c²-a²}}{x^{(a-b)+(b-c)+(c-a)}}$

$= \dfrac{x^0}{x^{a-b+b-c+c-a}}$

$= \dfrac{1}{x^0}$

$= \dfrac{1}{1}$

$= 1$
= Answer

Related Notes and Solutions:

Here is the website link to the notes of Indices.

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