Express as equivalent fractions with rational denominator:
$\frac{\sqrt{10} + \sqrt{5} - \sqrt{3}}{\sqrt{3} + \sqrt{10} - \sqrt{5}}$

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Solution

On rationalizing denominator, we get
$\frac{\sqrt{10} + (\sqrt{5} - \sqrt{3})}{\sqrt{10} - (\sqrt{5} - \sqrt{3})} \times \frac{\sqrt{10} + (\sqrt{5} - \sqrt{3})}{\sqrt{10} + (\sqrt{5} - \sqrt{3})}$
$= \frac{[\sqrt{10} + (\sqrt{5} - \sqrt{3})]^{2}}{(\sqrt{10})^{2} - (\sqrt{5} - \sqrt{3})^{2}}$
$= \frac{9 + 5 \sqrt{2} - \sqrt{15} - \sqrt{30}}{\sqrt{15} + 1} \times \frac{\sqrt{15} - 1}{\sqrt{15} - 1}$
$= \frac{3 \sqrt{30} + 5 \sqrt{15} - 10 \sqrt{2} - 12}{7}$

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