If a + b + c = 0, then $x^{a^{2} b^{- 1} c^{- 1}} . x^{a^{- 1} b^{- 1} c^{2}} . x^{b^{2} a^{- 1} c^{- 1}} =$ ___

x^{a^{2} b^{2} c^{2}}

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x^{1 / a^{2} b^{2} c^{2}}

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x^{1 / 2}

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x^{3}

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Solution

The correct option is D $x^{3}$
If $a + b + c = 0$ then
$x^{a^{2} b^{- 1} c^{- 1}} . x^{a^{- 1} b^{- 1} c^{2}} . x^{b^{2} a^{- 1} c^{- 1}}$
Property used : - $a^{- m} = \frac{1}{a^{m}}, a^{m} . a^{n} = a^{m + n}$
$∴ x^{a^{2} / b c} . x^{c^{2} / a b} . x^{b^{2} / a c}$
$= x \frac{a^{2}}{b c} + \frac{c^{2}}{a b} + \frac{b^{2}}{a b c}$
$= x \frac{a^{3}}{a b c} + \frac{b^{3}}{a b c} + \frac{c^{3}}{a b c}$ $∵ a + b + c = 0$
$= x + \frac{a^{3} + b^{3} + c^{3}}{a b c}$ $∵ a^{3} + b^{3} + c^{3} = 3 a b c$
$= x^{3}$

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