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

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Solution

On rationalizing denominator, we get
$\frac{(\sqrt{3} + (\sqrt{5}) (\sqrt{5} + \sqrt{2})}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$
$= \frac{\sqrt{15} + \sqrt{10} + \sqrt{6} + 5}{\sqrt{2} + \sqrt{3} + \sqrt{5}} \times \frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$
$= \frac{\sqrt{30} + 2 \sqrt{3} + 3 \sqrt{2}}{2 \sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}$
$= \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{2}$

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