Question

Express as equivalent fractions with rational denominator:
$\frac{\sqrt{2}}{\sqrt{2}+\sqrt{3}-\sqrt{5}}$

Answer 1

It can be written as $\frac{\sqrt{2}}{\sqrt{2}+\sqrt{3}-\sqrt{5}}$

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

$=\frac{2+\sqrt{6}+\sqrt{10}}{2+3+2\sqrt{6}-5}$

$=\frac{2+\sqrt{6}+\sqrt{10}}{2\sqrt{6}}$

$=\frac{2+\sqrt{6}+\sqrt{10}}{2\sqrt{6}}\times \frac{\sqrt{6}}{\sqrt{6}}$

$=\frac{2\sqrt{6}+6+\sqrt{60}}{2\times 6}$

$=\frac{\sqrt{6}+3\sqrt{15}}{6}$