# How do you rationalize the denominator and simplify  ((sqrt7) - (sqrt 2))/((sqrt7) + (sqrt 2))?

Apr 29, 2016

$\frac{9 - 2 \sqrt{14}}{5}$

#### Explanation:

Given:$\text{ } \textcolor{b r o w n}{\frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} + \sqrt{2}}}$

Trick is to use the scenario:$\text{ } {a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

Multiply by 1 but in the form $\textcolor{b l u e}{1 = \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}}$

$\textcolor{b r o w n}{\frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} + \sqrt{2}}} \textcolor{b l u e}{\times \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}}$

${\left(\sqrt{7} - \sqrt{2}\right)}^{2} / \left({\left(\sqrt{7}\right)}^{2} - {\left(\sqrt{2}\right)}^{2}\right)$

$\frac{7 - 2 \sqrt{7} \sqrt{2} + 2}{7 - 2}$

$\frac{9 - 2 \sqrt{14}}{5}$