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

Mar 16, 2018

The simplified expression is $- \frac{7 + 2 \sqrt{10}}{3}$.

#### Explanation:

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

$= \frac{\left(\sqrt{2} + \sqrt{5}\right)}{\left(\sqrt{2} - \sqrt{5}\right)} \textcolor{red}{\cdot \frac{\left(\sqrt{2} + \sqrt{5}\right)}{\left(\sqrt{2} + \sqrt{5}\right)}}$

$= \frac{\left(\sqrt{2} + \sqrt{5}\right) \left(\sqrt{2} + \sqrt{5}\right)}{\left(\sqrt{2} - \sqrt{5}\right) \left(\sqrt{2} + \sqrt{5}\right)}$

$= \frac{\left(\sqrt{2} + \sqrt{5}\right) \left(\sqrt{2} + \sqrt{5}\right)}{{\sqrt{2}}^{2} + \sqrt{2} \sqrt{5} - \sqrt{2} \sqrt{5} - {\sqrt{5}}^{2}}$

$= \frac{\left(\sqrt{2} + \sqrt{5}\right) \left(\sqrt{2} + \sqrt{5}\right)}{2 \textcolor{red}{\cancel{\textcolor{b l a c k}{+ \sqrt{10} - \sqrt{10}}}} - 5}$

$= \frac{\left(\sqrt{2} + \sqrt{5}\right) \left(\sqrt{2} + \sqrt{5}\right)}{2 - 5}$

$= \frac{\left(\sqrt{2} + \sqrt{5}\right) \left(\sqrt{2} + \sqrt{5}\right)}{- 3}$

$= \frac{{\sqrt{2}}^{2} + \sqrt{2} \sqrt{5} + \sqrt{2} \sqrt{5} + {\sqrt{5}}^{2}}{- 3}$

$= \frac{2 + \sqrt{10} + \sqrt{10} + 5}{- 3}$

$= \frac{2 + 2 \sqrt{10} + 5}{- 3}$

$= \frac{7 + 2 \sqrt{10}}{- 3}$

$= - \frac{7 + 2 \sqrt{10}}{3}$

This is the answer. You can verify using a calculator: