Question

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

Answer 1

On rationalizing denominator, we get

$\frac{(\sqrt{3}+(\sqrt{5}\left)\right(\sqrt{5}+\sqrt{2})}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$

$=\frac{\sqrt{15}+\sqrt{10}+\sqrt{6}+5}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\times \frac{\sqrt{2}+\sqrt{3}-\sqrt{5}}{\sqrt{2}+\sqrt{3}-\sqrt{5}}$

$=\frac{\sqrt{30}+2\sqrt{3}+3\sqrt{2}}{2\sqrt{6}}\times \frac{\sqrt{6}}{\sqrt{6}}$

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