Question

the following work models how to write the quotient \\(\frac{\sqrt3{4}}{\sqrt{2}}\\) using rational exponents.

\\\frac{\sqrt3{4}}{\sqrt{2}} = \frac{4^{\frac{1}{3}}}{2^{\frac{1}{2}}} = \frac{\left(2^2\
ight)^{\frac{1}{3}}}{2^{\frac{1}{2}}} = \frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}\\

note that, in order to use the quotient property of exponents, the bases must be the same.

using the work shown on the left, what is \\(\frac{\sqrt3{4}}{\sqrt{2}}\\) in simplest radical form?

\\(\bigcirc\\) 2
\\(\bigcirc\\) \\(\sqrt{2^6}\\)
\\(\bigcirc\\) \\(\sqrt6{2}\\)

Explanation:

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<reasoning>

Convert to rational exponents

Using the Radical to Exponential Form knowledge point
\[
\frac{\sqrt[3]{4}}{\sqrt{2}} = \frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}
\]

Apply quotient property of exponents

Using the Rational Exponents knowledge point
\[
2^{\frac{2}{3} - \frac{1}{2}} = 2^{\frac{4}{6} - \frac{3}{6}} = 2^{\frac{1}{6}}
\]

Convert back to radical form

Using the Radical to Exponential Form knowledge point
\[
2^{\frac{1}{6}} = \sqrt[6]{2}
\]
</reasoning>

<answer>
<mcq-option>2</mcq-option>
<mcq-option>\(\sqrt{2^6}\)</mcq-option>
<mcq-correct>\(\sqrt[6]{2}\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
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Answer:

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<reasoning>

Convert to rational exponents

Using the Radical to Exponential Form knowledge point
\[
\frac{\sqrt[3]{4}}{\sqrt{2}} = \frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}
\]

Apply quotient property of exponents

Using the Rational Exponents knowledge point
\[
2^{\frac{2}{3} - \frac{1}{2}} = 2^{\frac{4}{6} - \frac{3}{6}} = 2^{\frac{1}{6}}
\]

Convert back to radical form

Using the Radical to Exponential Form knowledge point
\[
2^{\frac{1}{6}} = \sqrt[6]{2}
\]
</reasoning>

<answer>
<mcq-option>2</mcq-option>
<mcq-option>\(\sqrt{2^6}\)</mcq-option>
<mcq-correct>\(\sqrt[6]{2}\)</mcq-correct>
</answer>

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