Radicals

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Rewrite the expression $\sqrt{x} \cdot \sqrt[3]{x}$ as a single radical expression.

$\sqrt[6]{x^{5}}$

CORRECT

$\sqrt[6]{x }$

$\sqrt[5]{x }$

$\sqrt[5]{x^{6} }$

$\sqrt[3]{x^{2}}$

First, rewrite each radical expression using rational exponents:

$\sqrt{x} = x ^{\frac{1}{2}}$

$\sqrt[3]{x} = x ^{\frac{1}{3}}$

Multiply these expressions using the Product of Powers Property:

$\sqrt{x} \cdot \sqrt[3]{x}$

$= x ^{\frac{1}{2}} \cdot x ^{\frac{1}{3}}$

$= x ^{\frac{1}{2}+\frac{1}{3}}$

$= x ^{\frac{3}{6}+\frac{2}{6}}$

$= x ^{\frac{5}{6}}$

This can be rewritten in radical form as $\sqrt[6]{x^{5}}$ .