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

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$\frac{1}{x^{3}}$

$x^{3}$

answer is D.

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Detailed Solution

If $a + b + c = 0$ , then $a^{3} + b^{3} + c^{3} = 3 a b c$

$= x^{a^{2} b^{- 1} c^{- 1}} . x^{a^{- 1} b^{2} c^{- 1}} . x^{a^{- 1} b^{- 1} c^{2}}$

$= x^{\frac{a^{2}}{b c}} . x^{\frac{b^{2}}{a c}} . x^{\frac{c^{2}}{a b}}$

$= x^{\frac{a^{2}}{b c} + \frac{b^{2}}{a c} + \frac{c^{2}}{a b}}$

$= x^{\frac{a^{3} + b^{3} + c^{3}}{a b c}}$

$= x^{\frac{3 a b c}{a b c}} = x^{3}$

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