Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
Simplify the following expressions.
$(\frac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+b}:(a-b)+\frac{2\sqrt{b}}{\sqrt{a}+\sqrt{b}})$

Answer 1

Given expression

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

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

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

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

$=\frac{a^2+b^2+\sqrt{ab}(a+b)}{a^2-b^{5/2}+2b^{3/2}+\sqrt{ab}|a-b+2|+a|\sqrt{a}+\sqrt{b}-1]-b^2\sqrt{2}}$