# How do you rationalize the denominator and simplify 5/(sqrt6+sqrt5)?

May 10, 2016

$\frac{5}{\sqrt{6} + \sqrt{5}} = 5 \sqrt{6} - 5 \sqrt{5}$

#### Explanation:

Use the difference of squares identity:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = \sqrt{6}$ and $b = \sqrt{5}$.

Multiply numerator and denominator by $\sqrt{6} - \sqrt{5}$

$\frac{5}{\sqrt{6} + \sqrt{5}}$

$= \frac{5 \left(\sqrt{6} - \sqrt{5}\right)}{\left(\sqrt{6} - \sqrt{5}\right) \left(\sqrt{6} + \sqrt{5}\right)}$

$= \frac{5 \left(\sqrt{6} - \sqrt{5}\right)}{6 - 5}$

$= \frac{5 \left(\sqrt{6} - \sqrt{5}\right)}{1}$

$= 5 \sqrt{6} - 5 \sqrt{5}$