# How do you rationalize the denominator and simplify 5/(sqrt3+sqrt5)?

Jun 4, 2017

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

#### Explanation:

If you have a denominator of type $\sqrt{a} + \sqrt{b}$, to rationalize multiply numerator and denominator by $\sqrt{a} - \sqrt{b}$ or vice-versa.

Hence to rationalize $\frac{5}{\sqrt{3} + \sqrt{5}}$ multiply numerator and denominator by $\sqrt{3} - \sqrt{5}$ and you get

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

= $\frac{5 \sqrt{3} - 5 \sqrt{5}}{3 - 5}$

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

= 5sqrt5-5sqrt3)/2