# How do you rationalize the denominator and simplify 8/(3-sqrt2)?

Mar 2, 2018

The answer is $\frac{24 + 8 \sqrt{2}}{7}$.

#### Explanation:

To rationalize the denominator of a fraction, multiply both the numerator and the denominator by the denominator's conjugate.

The conjugate of $3 - \sqrt{2}$ is $3 + \sqrt{2}$

$\textcolor{w h i t e}{=} \frac{8}{3 - \sqrt{2}}$

$= \frac{8}{\left(3 - \sqrt{2}\right)} \textcolor{red}{\cdot \frac{\left(3 + \sqrt{2}\right)}{\left(3 + \sqrt{2}\right)}}$

$= \frac{8 \left(3 + \sqrt{2}\right)}{\left(3 - \sqrt{2}\right) \left(3 + \sqrt{2}\right)}$

$= \frac{24 + 8 \sqrt{2}}{{3}^{2} + \textcolor{red}{\cancel{\textcolor{b l a c k}{3 \sqrt{2} - 3 \sqrt{2}}}} - {\left(\sqrt{2}\right)}^{2}}$

$= \frac{24 + 8 \sqrt{2}}{9 - 2}$

$= \frac{24 + 8 \sqrt{2}}{7}$

This answer is as simplified as possible.