Question

what is the following quotient?

\frac{9+\sqrt{2}}{4-\sqrt{7}}

\circ \frac{9\sqrt{7}+\sqrt{14}}{-3}

\circ \frac{36-9\sqrt{7}+4\sqrt{2}-\sqrt{14}}{9}

\circ \frac{36+9\sqrt{7}+4\sqrt{2}+\sqrt{14}}{9}

\circ \frac{79}{9}

Explanation:

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<reasoning>

Identify the conjugate of the denominator

Using the Radical Conjugates knowledge point
\[
\text{Conjugate of } 4 - \sqrt{7} \text{ is } 4 + \sqrt{7}
\]

Multiply the numerator and denominator by the conjugate

Using the Rationalizing the Denominator knowledge point
\[
\frac{9 + \sqrt{2}}{4 - \sqrt{7}} \cdot \frac{4 + \sqrt{7}}{4 + \sqrt{7}} = \frac{(9 + \sqrt{2})(4 + \sqrt{7})}{(4 - \sqrt{7})(4 + \sqrt{7})}
\]

Expand the numerator

Using the Multiplying Binomial Radicals knowledge point
\[
(9 + \sqrt{2})(4 + \sqrt{7}) = 9(4) + 9(\sqrt{7}) + \sqrt{2}(4) + \sqrt{2}(\sqrt{7}) = 36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}
\]

Simplify the denominator

Using the Radical Conjugates knowledge point
\[
(4 - \sqrt{7})(4 + \sqrt{7}) = 4^2 - (\sqrt{7})^2 = 16 - 7 = 9
\]

Write the final rationalized expression

Combine the simplified numerator and denominator:
\[
\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{9\sqrt{7} + \sqrt{14}}{-3}\)</mcq-option>
<mcq-option>(B) \(\frac{36 - 9\sqrt{7} + 4\sqrt{2} - \sqrt{14}}{9}\)</mcq-option>
<mcq-correct>(C) \(\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}\)</mcq-correct>
<mcq-option>(D) \(\frac{79}{9}\)</mcq-option>
</answer>

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"Mathematics",
"Algebra",
"Rationalizing the Denominator"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Identify the conjugate of the denominator

Using the Radical Conjugates knowledge point
\[
\text{Conjugate of } 4 - \sqrt{7} \text{ is } 4 + \sqrt{7}
\]

Multiply the numerator and denominator by the conjugate

Using the Rationalizing the Denominator knowledge point
\[
\frac{9 + \sqrt{2}}{4 - \sqrt{7}} \cdot \frac{4 + \sqrt{7}}{4 + \sqrt{7}} = \frac{(9 + \sqrt{2})(4 + \sqrt{7})}{(4 - \sqrt{7})(4 + \sqrt{7})}
\]

Expand the numerator

Using the Multiplying Binomial Radicals knowledge point
\[
(9 + \sqrt{2})(4 + \sqrt{7}) = 9(4) + 9(\sqrt{7}) + \sqrt{2}(4) + \sqrt{2}(\sqrt{7}) = 36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}
\]

Simplify the denominator

Using the Radical Conjugates knowledge point
\[
(4 - \sqrt{7})(4 + \sqrt{7}) = 4^2 - (\sqrt{7})^2 = 16 - 7 = 9
\]

Write the final rationalized expression

Combine the simplified numerator and denominator:
\[
\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{9\sqrt{7} + \sqrt{14}}{-3}\)</mcq-option>
<mcq-option>(B) \(\frac{36 - 9\sqrt{7} + 4\sqrt{2} - \sqrt{14}}{9}\)</mcq-option>
<mcq-correct>(C) \(\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}\)</mcq-correct>
<mcq-option>(D) \(\frac{79}{9}\)</mcq-option>
</answer>

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"Mathematics",
"Algebra",
"Rationalizing the Denominator"
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