Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{(5 \sqrt{3} + \sqrt{50}) (5 - \sqrt{24})}{\sqrt{75} - 5 \sqrt{2}}$

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Solution

$\frac{(5 \sqrt{3} + 5 \sqrt{2}) (5 - 2 \sqrt{6})}{(5 \sqrt{3} - 5 \sqrt{2})}$
$= \frac{5 (\sqrt{3} + \sqrt{2}) (5 - 2 \sqrt{6})}{5 (\sqrt{3} - \sqrt{2})}$
$= \frac{(\sqrt{3} + \sqrt{2}) (\sqrt{3} + \sqrt{2}) (5 - 2 \sqrt{6})}{(\sqrt{3} - \sqrt{2}) (\sqrt{3} + \sqrt{2})}$
$= \frac{(3 + 2 + 2 \sqrt{6}) (5 - 2 \sqrt{6})}{3 - 2}$
$= (5 + 2 \sqrt{6}) (5 - 2 \sqrt{6})$
$= 25 - 24 = 1$ .

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