Question

simplify the following radical expression by using the conjugate.

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

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

Explanation:

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

$$ \frac{\sqrt{13} - \sqrt{7}}{13 - 7} = \frac{\sqrt{13} - \sqrt{7}}{6} $$

Answer:

Simplify the following radical expression by using the conjugate.

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

$$\frac{\sqrt{13} - \sqrt{7}}{6}$$