Express as equivalent fractions with rational denominator:
$\frac{1}{1 + \sqrt{2} - \sqrt{3}}$

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Solution

It can be written as $\frac{1}{1 + \sqrt{2} - \sqrt{3}} \times \frac{1 + \sqrt{2} + \sqrt{3}}{1 + \sqrt{2} + \sqrt{3}}$
$= \frac{1 + \sqrt{2} + \sqrt{3}}{(1 + \sqrt{2})^{2} - (\sqrt{3})^{2}}$
$= \frac{1 + \sqrt{2} + \sqrt{3}}{2 \sqrt{2}}$
$= \frac{2 + \sqrt{2} + \sqrt{6}}{4}$

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