Express as equivalent fractions with rational denominator:
$\frac{2 \sqrt{a + 1}}{\sqrt{a - 1} - \sqrt{2 a} + \sqrt{a + 1}}$

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Solution

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

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