Question

simplify the following radical expression by using the conjugate.

\\\frac{1}{21 - \sqrt{11}}\\

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

Explanation:

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<reasoning>

Identify the conjugate of the denominator

Using the Rationalizing Denominators knowledge point
\[
\text{Denominator: } 21 - \sqrt{11}
\]
\[
\text{Conjugate: } 21 + \sqrt{11}
\]

Multiply numerator and denominator by the conjugate

Using the Rationalizing Denominators knowledge point
\[
\frac{1}{21 - \sqrt{11}} \cdot \frac{21 + \sqrt{11}}{21 + \sqrt{11}} = \frac{21 + \sqrt{11}}{(21 - \sqrt{11})(21 + \sqrt{11})}
\]

Simplify the denominator using difference of squares

Using the Rationalizing Denominators knowledge point
\[
(21 - \sqrt{11})(21 + \sqrt{11}) = 21^2 - (\sqrt{11})^2
\]
\[
21^2 - 11 = 441 - 11 = 430
\]

Write the final simplified expression

Using the Rationalizing Denominators knowledge point
\[
\frac{21 + \sqrt{11}}{430}
\]
</reasoning>

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

\[\frac{1}{21 - \sqrt{11}}\]

<blank>\(\frac{21 + \sqrt{11}}{430}\)</blank>
</answer>

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Answer:

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<reasoning>

Identify the conjugate of the denominator

Using the Rationalizing Denominators knowledge point
\[
\text{Denominator: } 21 - \sqrt{11}
\]
\[
\text{Conjugate: } 21 + \sqrt{11}
\]

Multiply numerator and denominator by the conjugate

Using the Rationalizing Denominators knowledge point
\[
\frac{1}{21 - \sqrt{11}} \cdot \frac{21 + \sqrt{11}}{21 + \sqrt{11}} = \frac{21 + \sqrt{11}}{(21 - \sqrt{11})(21 + \sqrt{11})}
\]

Simplify the denominator using difference of squares

Using the Rationalizing Denominators knowledge point
\[
(21 - \sqrt{11})(21 + \sqrt{11}) = 21^2 - (\sqrt{11})^2
\]
\[
21^2 - 11 = 441 - 11 = 430
\]

Write the final simplified expression

Using the Rationalizing Denominators knowledge point
\[
\frac{21 + \sqrt{11}}{430}
\]
</reasoning>

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

\[\frac{1}{21 - \sqrt{11}}\]

<blank>\(\frac{21 + \sqrt{11}}{430}\)</blank>
</answer>

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