Express as equivalent fractions with rational denominator:
$\frac{1}{\sqrt{a} + \sqrt{b} + \sqrt{a + b}}$

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Solution

On rationalizing denominator, we get
$\frac{1}{\sqrt{a} + \sqrt{b} + \sqrt{a + b}} \times \frac{\sqrt{a} + \sqrt{b} - \sqrt{a + b}}{\sqrt{a} + \sqrt{b} - \sqrt{a + b}}$
$= \frac{\sqrt{a} + \sqrt{b} - \sqrt{a + b}}{(\sqrt{a} + \sqrt{b})^{2} - (\sqrt{a + b})^{2}}$
$= \frac{\sqrt{a} + \sqrt{b} - \sqrt{a + b}}{2 \sqrt{a b}}$
$= \frac{a \sqrt{b} + b \sqrt{a} - \sqrt{a b (a + b)}}{2 a b}$

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