Question

simplify the following radical expression by using the conjugate.

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

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

Explanation:

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

$$ \frac{21 + \sqrt{11}}{430} $$

Answer:

Simplify the following radical expression by using the conjugate.

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

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