# How do you rationalize the denominator (1+sqrt2)/(3+sqrt5)?

May 28, 2015

Multiply both the numerator (top) and denominator (bottom) by the conjugate $3 - \sqrt{5}$...

$\frac{1 + \sqrt{2}}{3 + \sqrt{5}}$

$= \frac{1 + \sqrt{2}}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}}$

$= \frac{\left(1 + \sqrt{2}\right) \left(3 - \sqrt{5}\right)}{\left(3 + \sqrt{5}\right) \left(3 - \sqrt{5}\right)}$

$= \frac{\left(1 + \sqrt{2}\right) \left(3 - \sqrt{5}\right)}{{3}^{2} - {\sqrt{5}}^{2}}$

$= \frac{\left(1 + \sqrt{2}\right) \left(3 - \sqrt{5}\right)}{9 - 5}$

$= \frac{\left(1 + \sqrt{2}\right) \left(3 - \sqrt{5}\right)}{4}$