Represent the following mixed infinite decimal periodic fractions as common fractions:
$(\frac{1}{(a^{1 / 2} + b^{1 / 2})^{- 2}}) - {(\frac{\sqrt{a} - \sqrt{b}}{a^{3 / 2} - b^{3 / 2}})}^{- 1} (a b)^{- 1 / 2}$

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Solution

$[(\frac{1}{a^{\frac{1}{2}} + b^{\frac{1}{2}})^{- 2}}) - (\frac{\sqrt{a} - \sqrt{b}}{a^{\frac{3}{2}} - b^{\frac{3}{2}}})^{- 1}] (a b)^{\frac{- 1}{2}}$
$= [\sqrt{a} + \sqrt{b})^{2} - \frac{(\sqrt{a})^{3} - (\sqrt{b})^{3}}{\sqrt{a} - \sqrt{b}}] \sqrt{a b}$
$= [a + b + 2 \sqrt{a b} - \frac{(\sqrt{a} - \sqrt{b}) (a + b + \sqrt{a b})}{(\sqrt{a} - \sqrt{b})}] \sqrt{a b}$
$= (a + b + 2 \sqrt{a b} - a - b - \sqrt{a b})] \sqrt{a b}$
$= \sqrt{a b} \times \sqrt{a b}$
$= a b$

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