By what number should we multiply $(- 6)^{- 3}$ so that the product is ${(\frac{1}{3})}^{5}$

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Solution

Let the number should be multiplied is $x$

Given $x \times (- 6)^{- 3} = (\frac{1}{3})^{5}$

$⟹ \frac{x}{- 6^{3}} = \frac{1}{3^{5}}$

$⟹ x = \frac{- 6^{3}}{3^{5}} = - \frac{(2 \times 3)^{3}}{3^{5}} = - \frac{2^{3} \times 3^{3}}{3^{5}} = - \frac{2^{3}}{3^{2}} = - \frac{8}{9}$

So the number is $- \frac{8}{9}$

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