Question

which expression should you multiply the numerator and denominator of \\(\frac{\sqrt3{3}}{\sqrt3{2x}}\\) by to rationalize the denominator?

\\(\sqrt3{x^2}\\)
\\(\sqrt3{4x}\\)
\\(\sqrt3{4x^2}\\)
\\(\sqrt3{2x}\\)

what is \\(\frac{\sqrt3{3}}{\sqrt3{2x}}\\) in simplest form?

\\(\frac{\sqrt3{12x^2}}{2x}\\)
\\(\frac{\sqrt3{6x}}{2x}\\)
\\(\frac{\sqrt3{3}}{2x}\\)

Explanation:

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<reasoning>

Identify the rationalizing factor

Using the Rationalizing the Denominator knowledge point
\[

$$\begin{aligned} &\text{Denominator: } \sqrt[3]{2x} = \sqrt[3]{2^1 \cdot x^1}\\ &\text{To make the radicand a perfect cube, multiply by: } \sqrt[3]{2^{3-1} \cdot x^{3-1}} = \sqrt[3]{2^2 \cdot x^2} = \sqrt[3]{4x^2} \end{aligned}$$

\]

Simplify the expression

Using the Rationalizing the Denominator knowledge point
\[

$$\begin{aligned} \frac{\sqrt[3]{3}}{\sqrt[3]{2x}} \cdot \frac{\sqrt[3]{4x^2}}{\sqrt[3]{4x^2}} &= \frac{\sqrt[3]{3 \cdot 4x^2}}{\sqrt[3]{2x \cdot 4x^2}} \\ &= \frac{\sqrt[3]{12x^2}}{\sqrt[3]{8x^3}} \\ &= \frac{\sqrt[3]{12x^2}}{2x} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(\sqrt[3]{x^2}\)</mcq-option>
<mcq-option>(B) \(\sqrt[3]{4x}\)</mcq-option>
<mcq-correct>(C) \(\sqrt[3]{4x^2}\)</mcq-correct>
<mcq-option>(D) \(\sqrt[3]{2x}\)</mcq-option>

Question 2

<mcq-correct>(A) \(\frac{\sqrt[3]{12x^2}}{2x}\)</mcq-correct>
<mcq-option>(B) \(\frac{\sqrt[3]{6x}}{2x}\)</mcq-option>
<mcq-option>(C) \(\frac{\sqrt[3]{3}}{2x}\)</mcq-option>
</answer>

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</post_analysis>

Answer:

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<reasoning>

Identify the rationalizing factor

Using the Rationalizing the Denominator knowledge point
\[

$$\begin{aligned} &\text{Denominator: } \sqrt[3]{2x} = \sqrt[3]{2^1 \cdot x^1}\\ &\text{To make the radicand a perfect cube, multiply by: } \sqrt[3]{2^{3-1} \cdot x^{3-1}} = \sqrt[3]{2^2 \cdot x^2} = \sqrt[3]{4x^2} \end{aligned}$$

\]

Simplify the expression

Using the Rationalizing the Denominator knowledge point
\[

$$\begin{aligned} \frac{\sqrt[3]{3}}{\sqrt[3]{2x}} \cdot \frac{\sqrt[3]{4x^2}}{\sqrt[3]{4x^2}} &= \frac{\sqrt[3]{3 \cdot 4x^2}}{\sqrt[3]{2x \cdot 4x^2}} \\ &= \frac{\sqrt[3]{12x^2}}{\sqrt[3]{8x^3}} \\ &= \frac{\sqrt[3]{12x^2}}{2x} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(\sqrt[3]{x^2}\)</mcq-option>
<mcq-option>(B) \(\sqrt[3]{4x}\)</mcq-option>
<mcq-correct>(C) \(\sqrt[3]{4x^2}\)</mcq-correct>
<mcq-option>(D) \(\sqrt[3]{2x}\)</mcq-option>

Question 2

<mcq-correct>(A) \(\frac{\sqrt[3]{12x^2}}{2x}\)</mcq-correct>
<mcq-option>(B) \(\frac{\sqrt[3]{6x}}{2x}\)</mcq-option>
<mcq-option>(C) \(\frac{\sqrt[3]{3}}{2x}\)</mcq-option>
</answer>

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