Question

simplify the following radical expression by using the conjugate.

\\\frac{1}{\sqrt{13} + \sqrt{7}}\\

\\\frac{\sqrt{?} - \sqrt{7}}{ }\\

Explanation:

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<reasoning>

Identify the conjugate of the denominator

\[
\text{Denominator} = \sqrt{13} + \sqrt{7} \implies \text{Conjugate} = \sqrt{13} - \sqrt{7}
\]

Multiply numerator and denominator by the conjugate

\[
\frac{1}{\sqrt{13} + \sqrt{7}} \cdot \frac{\sqrt{13} - \sqrt{7}}{\sqrt{13} - \sqrt{7}} = \frac{\sqrt{13} - \sqrt{7}}{(\sqrt{13})^2 - (\sqrt{7})^2}
\]

Simplify the denominator

\[
\frac{\sqrt{13} - \sqrt{7}}{13 - 7} = \frac{\sqrt{13} - \sqrt{7}}{6}
\]
</reasoning>

<answer>
Simplify the following radical expression by using the conjugate.

\[\frac{1}{\sqrt{13} + \sqrt{7}}\]

\[\frac{\sqrt{13} - \sqrt{7}}{6}\]
</answer>

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"Mathematics",
"Algebra",
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Answer:

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<reasoning>

Identify the conjugate of the denominator

\[
\text{Denominator} = \sqrt{13} + \sqrt{7} \implies \text{Conjugate} = \sqrt{13} - \sqrt{7}
\]

Multiply numerator and denominator by the conjugate

\[
\frac{1}{\sqrt{13} + \sqrt{7}} \cdot \frac{\sqrt{13} - \sqrt{7}}{\sqrt{13} - \sqrt{7}} = \frac{\sqrt{13} - \sqrt{7}}{(\sqrt{13})^2 - (\sqrt{7})^2}
\]

Simplify the denominator

\[
\frac{\sqrt{13} - \sqrt{7}}{13 - 7} = \frac{\sqrt{13} - \sqrt{7}}{6}
\]
</reasoning>

<answer>
Simplify the following radical expression by using the conjugate.

\[\frac{1}{\sqrt{13} + \sqrt{7}}\]

\[\frac{\sqrt{13} - \sqrt{7}}{6}\]
</answer>

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