Question

to rationalize the denominator of \\(\frac{5 - \sqrt{7}}{9 - \sqrt{14}}\\), you should multiply the expression by which fraction?

\\(\circ\\) \\(\frac{5 + \sqrt{7}}{9 - \sqrt{14}}\\)

\\(\circ\\) \\(\frac{9 - \sqrt{14}}{9 - \sqrt{14}}\\)

\\(\circ\\) \\(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\\)

\\(\circ\\) \\(\frac{\sqrt{14}}{\sqrt{14}}\\)

Explanation:

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<reasoning>

Identify the denominator and its conjugate

Using the Radical Conjugates knowledge point
\[

$$\begin{aligned} &\text{Denominator: } 9 - \sqrt{14}\\ &\text{Conjugate: } 9 + \sqrt{14} \end{aligned}$$

\]

Determine the rationalizing fraction

Using the Rationalizing the Denominator knowledge point
\[
\frac{9 + \sqrt{14}}{9 + \sqrt{14}}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{5 + \sqrt{7}}{9 - \sqrt{14}}\)</mcq-option>
<mcq-option>(B) \(\frac{9 - \sqrt{14}}{9 - \sqrt{14}}\)</mcq-option>
<mcq-correct>(C) \(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\)</mcq-correct>
<mcq-option>(D) \(\frac{\sqrt{14}}{\sqrt{14}}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the denominator and its conjugate

Using the Radical Conjugates knowledge point
\[

$$\begin{aligned} &\text{Denominator: } 9 - \sqrt{14}\\ &\text{Conjugate: } 9 + \sqrt{14} \end{aligned}$$

\]

Determine the rationalizing fraction

Using the Rationalizing the Denominator knowledge point
\[
\frac{9 + \sqrt{14}}{9 + \sqrt{14}}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{5 + \sqrt{7}}{9 - \sqrt{14}}\)</mcq-option>
<mcq-option>(B) \(\frac{9 - \sqrt{14}}{9 - \sqrt{14}}\)</mcq-option>
<mcq-correct>(C) \(\frac{9 + \sqrt{14}}{9 + \sqrt{14}}\)</mcq-correct>
<mcq-option>(D) \(\frac{\sqrt{14}}{\sqrt{14}}\)</mcq-option>
</answer>

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