Represent the following mixed infinite decimal periodic fractions as common fractions:
$2 (a + b)^{- 1} (a b)^{1 / 2} {⎛ ⎝ 1 + \frac{1}{4} {(\sqrt{\frac{a}{b}} - \sqrt{\frac{b}{a}})}^{2} ⎞ ⎠}^{1 / 2}$

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Solution

$2 (a + b)^{- 1} (a b)^{\frac{1}{2}} (1 + \frac{1}{4} (\sqrt{\frac{a}{b}} - \sqrt{\frac{b}{a}})^{2})^{\frac{1}{2}}$
$= 2 (a + b)^{- 1} \sqrt{a b} \frac{(4 a b + (a - b)^{2})^{\frac{1}{2}}}{(4 a b)^{\frac{1}{2}}}$
$= 2 (a + b)^{- 1} \sqrt{a b} \frac{((a + b)^{2})^{\frac{1}{2}}}{2 \sqrt{a b}}$
$= (a + b)^{- 1} (a + b) = 1$

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