Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{\sqrt{x} + 1}{1 + \sqrt{x} + x} : \frac{1}{x^{2} - \sqrt{x}}$

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Solution

$\frac{\sqrt{x} + 1}{1 + \sqrt{x} + x} : \frac{1}{x^{2} - \sqrt{x}}$
$= \frac{(\sqrt{x} + 1) (\sqrt{x} - 1)}{(1 + \sqrt{x} + x) (\sqrt{x} - 1)} : \frac{1}{\sqrt{x} (x^{2 - \frac{1}{2}} - 1)}$
$= \frac{(\sqrt{x})^{2} - 1}{(\sqrt{x})^{3} - 1} \times \sqrt{x} (x^{\frac{3}{2}} - 1)$
$= \frac{(x - 1)}{x^{\frac{3}{2}} - 1} \sqrt{x} (x^{\frac{3}{2}} - 1)$
$= \sqrt{x} (x - 1)$

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