Question

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

Answer 1

Given expression

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

$=(a^2\sqrt{b}{)}^{-\frac{1}{2}}\left[\frac{(a+\sqrt{ab})\left(\sqrt{a}b\right)-ab}{a+\sqrt{ab}}\right]:\frac{\sqrt[4]{4ab}-\sqrt{b}}{a-b}$

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

$={\frac{1}{\left(a^2\sqrt{b}\right)}}^{1/2}=\frac{a\sqrt{ab}}{a{b}^{1/4}(a+\sqrt{ab})}=\frac{{\sqrt{ab}}^{1/4}(a-b)}{(a+\sqrt{ab})(\sqrt[4]{4ab}-\sqrt{b})}$

$=\frac{\sqrt[4]{4ab}-\sqrt{b}}{a-b}=\frac{\sqrt{a}(a-b)}{(a+\sqrt{ab})(\sqrt[4]{4a}-\sqrt[4]{4b})}$

$=\frac{\sqrt{a}(a-b)}{\sqrt{a}(\sqrt{a}+\sqrt{b})(\sqrt[a]{aa}-\sqrt[a]{ab})}=\frac{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}{\left(\sqrt{a+\sqrt{b}}\right)(\sqrt[4]{4a}-\sqrt[4]{4b})}$

$=\frac{(\sqrt[4]{4a}+\sqrt[4]{4b})(\sqrt[4]{4b}-\sqrt[4]{4b})}{(\sqrt[4]{4a}-\sqrt[4]{4b})}$

$=\sqrt[4]{4a}+\sqrt[4]{4b}$