Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{b - x}{\sqrt{b} - \sqrt{x}} - \frac{b^{3 / 2} - x^{3 / 2}}{b - x}$

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Solution

$\frac{b - x}{\sqrt{b} - \sqrt{x}} - \frac{b^{\frac{3}{2}} - x^{\frac{3}{2}}}{b - x}$
$= \frac{(\sqrt{b})^{2} - (\sqrt{x})^{2}}{\sqrt{b} - \sqrt{x}} - \frac{(\sqrt{b} - \sqrt{x}) (b + x + \sqrt{b} x)}{(\sqrt{b})^{2} - (\sqrt{x})^{2}}$
$= \sqrt{b} + \sqrt{x} - \frac{(b + x - \sqrt{b x})}{\sqrt{b} + \sqrt{x}}$
$= \frac{b + x + 2 \sqrt{b x} - b - x - \sqrt{b x}}{\sqrt{b} + \sqrt{x}}$
$= \frac{\sqrt{b x}}{\sqrt{b} + \sqrt{x}}$

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