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


Solution:
Given,

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

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

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

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

$= x^{a²-b²} × x^{b²-c²} × x^{c²-a²}$

$= x^{(a²-b²) +(b²-c²) +c²-a²)}$

$= x^{a²-b²+b²-c²+c²-a²}$

$= x^0$

$= 1$
= Answer

Related Notes and Solutions:

Here is the website link to the notes of Indices.

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