If ${(6 x)}^{6} = 6^{2^{3}}, then x =________.$

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Solution

$Let {(6 x)}^{6} = 6^{2^{3}} \Rightarrow {(6 x)}^{6} = 6^{8} \Rightarrow 6^{6} x^{6} = 6^{8} \Rightarrow x^{6} = \frac{6^{8}}{6^{6}} \Rightarrow x^{6} = 6^{8 - 6} \Rightarrow x^{6} = 6^{2} \Rightarrow x = {(6^{2})}^{\frac{1}{6}} \Rightarrow x = 6^{2 \times \frac{1}{6}} \Rightarrow x = 6^{\frac{1}{3}}$

Hence, if ${(6 x)}^{6} = 6^{2^{3}}, then x = 6^{\frac{1}{3}} .$

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