If $a = x^{1 / 3} - x^{- 1 / 3}$ , then $a^{3} + 3 a =$ ___

x - x^{- 1}

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x^{- 1} - x

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x + x^{- 1}

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x^{- 1} + x

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Solution

The correct option is A $x - x^{- 1}$
$a = x^{1 / 3} - x^{- 1 / 3}$
then $a^{3} + 3 a = (x^{1 / 3} - x^{- 1 / 3})^{3} + 3 (x^{1 / 3} - x^{- 1 / 3})$
$\Rightarrow x - x^{- 1} 3 x^{2 / 3} x^{- 1 / 3} + 3 x^{1 / 3} x^{- 2 / 3} + 3 x^{1 / 3} - 3 x^{- 1 / 3}$
$\Rightarrow x - x^{- 1} - 3 x^{1 / 3} + 3 x^{- 1 / 3} + 3 x^{1 / 3} - 3 x^{- 1 / 3}$
$\Rightarrow x - x^{- 1}$

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