{x^(b/c)/x^(c/b)}^(1/bc) * {x^(c/a)/x^(a/c)}^(1/ca) * {x^(a/b)/x^(b/a)}^(1/ab) | Indices - Simplify | SciPiPupil

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

Solution:

Given,

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

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

$= \left ( x^{\frac{b² -c²}{bc}} \right ) ^{\frac{1}{bc}} × \left ( x^{\frac{c²-a²}{ca}} \right ) ^{\frac{1}{ca}} × \left ( x^{\frac{a²-b²}{ab}} \right ) ^{\frac{1}{ab}}$

$= \left ( x^{\frac{b²-c²}{bc}} \right ) ^{\frac{1}{bc}} × \left ( x^{\frac{c²-a²}{ca}} \right ) ^{\frac{1}{ca}} × \left ( x^{\frac{a²-b²}{ab}} \right ) ^{\frac{1}{ab}}$

$= x^{b²-c²} × x^{c²-a²} × x^{a² -b²}$

$= x^{(b²-c²)+(c²-a²) +(a²-b²)}$

$= x^{b²-c²+c²-a²+a²-b²}$

$= x^0$

$= 1$

= Answer

Related Notes and Solutions:

Here is the website link to the notes of Indices.

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