Question

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

Answer 1

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

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

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

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

$=\frac{2\sqrt{b}+\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}$

$=\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}}=1$