# How do you rationalize the denominator and simplify (-sqrt3)/ (sqrt 2 + 5)?

Oct 8, 2017

See a solution process below:

#### Explanation:

To rationalize the denominator we need to multiply it by the appropriate form of $1$. For this form of denominator we use this rule of quadratics to determine what to multiply by:

$\left(\textcolor{red}{x} + \textcolor{b l u e}{y}\right) \left(\textcolor{red}{x} - \textcolor{b l u e}{y}\right) = {\textcolor{red}{x}}^{2} - {\textcolor{b l u e}{y}}^{2}$

$\frac{\sqrt{\textcolor{red}{2}} - \textcolor{b l u e}{5}}{\sqrt{\textcolor{red}{2}} - \textcolor{b l u e}{5}} \times \frac{- 3}{\sqrt{\textcolor{red}{2}} + \textcolor{b l u e}{5}} \implies$

$\frac{- 3 \left(\sqrt{\textcolor{red}{2}} - \textcolor{b l u e}{5}\right)}{{\sqrt{\textcolor{red}{2}}}^{2} - {\textcolor{b l u e}{5}}^{2}} \implies$

$\frac{\left(- 3 \times \sqrt{\textcolor{red}{2}}\right) - \left(- 3 \times \textcolor{b l u e}{5}\right)}{2 - 25} \implies$

$\frac{- 3 \sqrt{2} - \left(- 15\right)}{2 - 25} \implies$

$\frac{- 3 \sqrt{2} + 15}{- 23} \implies$

$- \frac{15 - 3 \sqrt{2}}{23}$