Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
Simplify the following expressions.
$\frac{b^{1/2}}{a^{1/2}+1}:(\frac{\sqrt{b}\frac{a}{(ab{)}^{-0.5}}}{1-a}-\sqrt{ab})+\frac{b}{a}{(-3\frac{3}{8})}^{-1/3}$

Answer 1

Given expression

$\frac{b^{1/2}}{a^{1/2}+1}:(\frac{\sqrt{b}\frac{a}{(ab{)}^{-0.5}}}{1-a}-\sqrt{ab})+\frac{b}{a}{(-3\frac{3}{8})}^{-1/3}$

$=\frac{\sqrt{b}}{\sqrt{a}+1}(\frac{a\sqrt{b}(ab{)}^{+0.5}}{1-a}-\sqrt{ab})-\frac{b}{a}{\left(\frac{-9}{8}\right)}^{-1/3}$

$=\frac{\sqrt{b}}{\sqrt{a}+1}(\frac{a\sqrt{b}\sqrt{ab}}{1-a}-\sqrt{ab})+\frac{1}{a}{\left(\frac{-8}{9}\right)}^{1/3}$

$=\frac{\sqrt{b}\sqrt{ab}}{\sqrt{a+1}}(\frac{a\sqrt{b}}{1-a}-1)+\frac{b}{a}\frac{(-1{)}^{1/3}(8{)}^{1/3}}{(a{)}^{1/3}}$

$=\frac{\sqrt{ab}}{\sqrt{a}+1}\left(\frac{a\sqrt{b}-1+a}{1-a}\right)+\frac{1}{a}\frac{(-1)\left(2\right)}{\sqrt[3]{39}}$

$=\frac{ab\sqrt{ab}-b\sqrt{a}+ab\sqrt{a}}{1-a+\sqrt{a}-a\sqrt{a}}-\frac{2b}{a\sqrt[3]{39}}$

$=\frac{ab\left(a\sqrt{ab}{a}^{3/2}b\right(\sqrt{b}-b\sqrt{b}\left)\right)}{1-a+\sqrt{a}-a\sqrt{a}}-\frac{2b}{a\sqrt[3]{39}}$

$=\frac{a^{3/2}{b}^{3/2}|1-\sqrt{ab}|}{(1-a)(1+\sqrt{a})}-\frac{2b}{a\sqrt[3]{39}}$