Question

simplify the following radical expression by using the conjugate.

\\\frac{1}{\sqrt{14} - \sqrt{3}}\\

\\\frac{\sqrt{?} + \sqrt{3}}{\quad}\\

Explanation:

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<reasoning>

Identify the conjugate of the denominator

\[
\text{Denominator: } \sqrt{14} - \sqrt{3} \implies \text{Conjugate: } \sqrt{14} + \sqrt{3}
\]

Multiply the numerator and denominator by the conjugate

\[
\frac{1}{\sqrt{14} - \sqrt{3}} \cdot \frac{\sqrt{14} + \sqrt{3}}{\sqrt{14} + \sqrt{3}} = \frac{\sqrt{14} + \sqrt{3}}{(\sqrt{14})^2 - (\sqrt{3})^2}
\]

Simplify the denominator and match the target expression

\[
\frac{\sqrt{14} + \sqrt{3}}{14 - 3} = \frac{\sqrt{14} + \sqrt{3}}{11}
\]
</reasoning>

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

\[\frac{1}{\sqrt{14} - \sqrt{3}}\]

\[\frac{\sqrt{\text{14}} + \sqrt{3}}{\text{11}}\]
</answer>

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"Mathematics",
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Answer:

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<reasoning>

Identify the conjugate of the denominator

\[
\text{Denominator: } \sqrt{14} - \sqrt{3} \implies \text{Conjugate: } \sqrt{14} + \sqrt{3}
\]

Multiply the numerator and denominator by the conjugate

\[
\frac{1}{\sqrt{14} - \sqrt{3}} \cdot \frac{\sqrt{14} + \sqrt{3}}{\sqrt{14} + \sqrt{3}} = \frac{\sqrt{14} + \sqrt{3}}{(\sqrt{14})^2 - (\sqrt{3})^2}
\]

Simplify the denominator and match the target expression

\[
\frac{\sqrt{14} + \sqrt{3}}{14 - 3} = \frac{\sqrt{14} + \sqrt{3}}{11}
\]
</reasoning>

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

\[\frac{1}{\sqrt{14} - \sqrt{3}}\]

\[\frac{\sqrt{\text{14}} + \sqrt{3}}{\text{11}}\]
</answer>

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