Simplify and express the result in its simplest form

$\sqrt[6]{24} \div (\sqrt[3]{3} \cdot \sqrt{2})$

\frac{1}{\sqrt[6]{3}}

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\frac{1}{\sqrt[3]{3}}

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\frac{1}{\sqrt[6]{2}}

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\frac{1}{\sqrt[6]{4}}

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Solution

The correct option is D $\frac{1}{\sqrt[6]{3}}$
$\sqrt[6]{24} \div (\sqrt[3]{3} \cdot \sqrt{2})$

$= \sqrt[6]{24} \div (3^{\frac{1}{3}} \cdot 2^{\frac{1}{2}})$

$= \sqrt[6]{24} \div (3^{\frac{1 \times 2}{3 \times 2}} \cdot 2^{\frac{1 \times 3}{2 \times 3}})$

$= \sqrt[6]{24} \div \sqrt[6]{9 \cdot 8}$

$= \sqrt[6]{24 \div 72}$

$= \frac{1}{\sqrt[6]{3}}$

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