Question

which of the following is equivalent to \frac{\sqrt3{32x^3y^6}}{\sqrt3{2x^9y^2}}, where \\(x \ge 0\\) and \\(y \ge 0\\)?

• \\(\sqrt3{16x^6y^4}\\)
• \\(\sqrt3{\frac{y^4}{16x^6}}\\)
• \\(\sqrt3{\frac{16y^4}{x^6}}\\)

Explanation:

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<reasoning>

Apply quotient property of radicals

Using the Algebraic Simplification knowledge point
\[
\frac{\sqrt[3]{32x^3y^6}}{\sqrt[3]{2x^9y^2}} = \sqrt[3]{\frac{32x^3y^6}{2x^9y^2}}
\]

Simplify the fraction inside the radical

Using the Exponent Rules knowledge point
\[

$$\begin{aligned} \frac{32}{2} &= 16 \\ \frac{x^3}{x^9} &= x^{3-9} = x^{-6} = \frac{1}{x^6} \\ \frac{y^6}{y^2} &= y^{6-2} = y^4 \\ \frac{32x^3y^6}{2x^9y^2} &= \frac{16y^4}{x^6} \end{aligned}$$

\]

Write the final simplified radical expression

Using the Algebraic Simplification knowledge point
\[
\sqrt[3]{\frac{16y^4}{x^6}}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\sqrt[3]{16x^6y^4}\)</mcq-option>
<mcq-option>(B) \(\sqrt[3]{\frac{y^4}{16x^6}}\)</mcq-option>
<mcq-correct>(C) \(\sqrt[3]{\frac{16y^4}{x^6}}\)</mcq-correct>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Apply quotient property of radicals

Using the Algebraic Simplification knowledge point
\[
\frac{\sqrt[3]{32x^3y^6}}{\sqrt[3]{2x^9y^2}} = \sqrt[3]{\frac{32x^3y^6}{2x^9y^2}}
\]

Simplify the fraction inside the radical

Using the Exponent Rules knowledge point
\[

$$\begin{aligned} \frac{32}{2} &= 16 \\ \frac{x^3}{x^9} &= x^{3-9} = x^{-6} = \frac{1}{x^6} \\ \frac{y^6}{y^2} &= y^{6-2} = y^4 \\ \frac{32x^3y^6}{2x^9y^2} &= \frac{16y^4}{x^6} \end{aligned}$$

\]

Write the final simplified radical expression

Using the Algebraic Simplification knowledge point
\[
\sqrt[3]{\frac{16y^4}{x^6}}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\sqrt[3]{16x^6y^4}\)</mcq-option>
<mcq-option>(B) \(\sqrt[3]{\frac{y^4}{16x^6}}\)</mcq-option>
<mcq-correct>(C) \(\sqrt[3]{\frac{16y^4}{x^6}}\)</mcq-correct>
</answer>

<post_analysis>
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