# How do you simplify 2 / (5 - sqrt 2)?

Oct 24, 2015

$\frac{2}{5 - \sqrt{2}} = \frac{10 + 2 \sqrt{2}}{23}$

#### Explanation:

We can rationalize the denominator by multiplying it (and the numerator) by the conjugate of the denominator.

$\frac{2}{5 - \sqrt{2}}$
$\textcolor{w h i t e}{\text{XXX}} = \frac{2}{5 - \sqrt{2}} \times \frac{5 + \sqrt{2}}{5 + \sqrt{2}}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{10 + 2 \sqrt{2}}{{5}^{2} - {\left(\sqrt{2}\right)}^{2}}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{10 + 2 \sqrt{2}}{23}$

(Some people would consider this "simpler").