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

Jan 18, 2017

$\sqrt{15} - 4$

Explanation:

You would use the formula:

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

to razionalize the denominator, then

$\frac{\sqrt{3} - \sqrt{5}}{\sqrt{5} + \sqrt{3}} = \frac{\left(\sqrt{3} - \sqrt{5}\right) \left(\sqrt{5} - \sqrt{3}\right)}{\left(\sqrt{5} + \sqrt{3}\right) \left(\sqrt{5} - \sqrt{3}\right)} = \textcolor{red}{-} \frac{{\left(\sqrt{3} - \sqrt{5}\right)}^{2}}{5 - 3} = - \frac{3 + 5 - 2 \sqrt{15}}{2} = \frac{2 \sqrt{15} - 8}{2} = \sqrt{15} - 4$