Question

Simplify each of the following by rationalising the denominator:
$\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}$

Answer 1

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

Multiply by $(\sqrt{7}-\sqrt{5})$ numerator and denominator we get

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

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

=$\frac{7-2\sqrt{35}-5}{7-5}$

=$\frac{2-2\sqrt{35}}{2}$

=$1-\sqrt{35}$