# How do you simplify (sqrta- sqrtb)/(sqrta+sqrtb)?

May 19, 2015

Multiply both numerator and denominator by $\sqrt{a} - \sqrt{b}$

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

May 19, 2015

In this case, simplifying means rationalizing the denominator, that is getting rid of its square roots.

This can be done by using a well known algebraic identity: ${x}^{2} - {y}^{2} = \left(x + y\right) \left(x - y\right)$.

In our case, we have to multiply the fraction by $\left(\sqrt{a} - \sqrt{b}\right)$:

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