Question

Simplify the following by rationalizing the denominator.
$\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$

Answer 1

Given

$\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$

$=\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\times \frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}-\sqrt{5}}$

$=\frac{(\sqrt{7}-\sqrt{5}{)}^2}{(\sqrt{7}{)}^2-(\sqrt{5}{)}^2}$

$=\frac{7+5-2\left(\sqrt{7}\right)\left(\sqrt{5}\right)}{(\sqrt{7}{)}^2-(\sqrt{5}{)}^2}$

$=\frac{12-2\left(\sqrt{35}\right)}{7-5}$

$=\frac{2(6-\sqrt{35})}{2}$

$=6-\sqrt{35}$