How do you rationalize the denominator and simplify 1 / (sqrt5+sqrt6)?

Mar 16, 2016

$\frac{1}{\sqrt{5} + \sqrt{6}} = \sqrt{6} - \sqrt{5}$

Explanation:

It works out a little easier if you swap $\sqrt{5} + \sqrt{6}$ around first.

Multiply both numerator and denominator by the conjugate $\sqrt{6} - \sqrt{5}$ of the denominator:

$\frac{1}{\sqrt{5} + \sqrt{6}}$

$= \frac{1}{\sqrt{6} + \sqrt{5}}$

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

$= \frac{\sqrt{6} - \sqrt{5}}{{\left(\sqrt{6}\right)}^{2} - {\left(\sqrt{5}\right)}^{2}}$

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

$= \frac{\sqrt{6} - \sqrt{5}}{1}$

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