# How do you rationalize the denominator and simplify 4/(6-sqrt5)?

Oct 11, 2015

Multiply both numerator and denominator by $\left(6 + \sqrt{5}\right)$ and simplify to find:

$\frac{4}{6 - \sqrt{5}} = \frac{24 + 4 \sqrt{5}}{31}$

#### Explanation:

$\frac{4}{6 - \sqrt{5}} = \frac{4 \left(6 + \sqrt{5}\right)}{\left(6 - \sqrt{5}\right) \left(6 + \sqrt{5}\right)} =$

$= \frac{24 + 4 \sqrt{5}}{{6}^{2} - {\sqrt{5}}^{2}} = \frac{24 + 4 \sqrt{5}}{36 - 5}$

$= \frac{24 + 4 \sqrt{5}}{31}$

This uses the difference of squares identity to square any square roots in the binomial denominator.

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

In our case $a = 6$ and $b = \sqrt{5}$, but it would work if both $a$ and $b$ were square roots.