Question

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

Answer 1

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

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

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

$=(a+b{)}^{-1}(a+b)=1$