Question

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

Answer 1

$\left[\right(\frac{1}{a^{\frac{1}{2}}+b^{\frac{1}{2}}{)}^{-2}})-(\frac{\sqrt{a}-\sqrt{b}}{a^{\frac{3}{2}}-b^{\frac{3}{2}}}{)}^{-1}\left]\right(ab{)}^{\frac{-1}{2}}$

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

$=[a+b+2\sqrt{ab}-\frac{(\sqrt{a}-\sqrt{b})(a+b+\sqrt{ab})}{(\sqrt{a}-\sqrt{b})}]\sqrt{ab}$

$=(a+b+2\sqrt{ab}-a-b-\sqrt{ab})]\sqrt{ab}$

$=\sqrt{ab}\times \sqrt{ab}$

$=ab$