Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{3\frac{1}{3}\sqrt{\frac{9}{80}}-\frac{5}{4}\sqrt{\frac{4}{5}}+5\sqrt{\frac{1}{5}}+\sqrt{20}-10\sqrt{0.2}}{3\frac{1}{2}\sqrt{32}-\sqrt{4\frac{1}{2}}+2\sqrt{\frac{1}{8}}+6\sqrt{\frac{2}{9}}-140\sqrt{0.02}}\sqrt{\frac{2}{3}}$

Answer 1

$\frac{3\frac{1}{3}\sqrt{\frac{9}{80}}-\frac{5}{4}\sqrt{\frac{4}{5}}+5\sqrt{\frac{1}{5}}+\sqrt{20}-10\sqrt{0.2}}{3\frac{1}{2}\sqrt{32}-\sqrt{4\frac{1}{2}}+2\sqrt{\frac{1}{8}}+6\sqrt{\frac{2}{9}}-140\sqrt{0.02}}\sqrt{\frac{2}{3}}$

$=\frac{\frac{10}{3}\frac{3}{4\sqrt{5}}-\frac{5}{4}.\frac{2}{\sqrt{5}}+\frac{5}{\sqrt{5}}+2\sqrt{5}-\frac{10}{\sqrt{5}}}{\frac{7}{2}.4\sqrt{2}-\frac{3}{\sqrt{2}}+\frac{1}{\sqrt{2}}.\frac{6\sqrt{2}}{3}-\frac{140}{5\sqrt{2}}}$ $\sqrt{\frac{2}{3}}$

$=\frac{\sqrt{5}-\sqrt{5}+\sqrt{5}+2\sqrt{5}-2\sqrt{5}}{\frac{28}{\sqrt{2}}-\frac{3}{\sqrt{2}}+\frac{1}{\sqrt{2}}+\frac{4}{\sqrt{2}}-\frac{28}{\sqrt{2}}}$ $\sqrt{\frac{2}{3}}$

$=\frac{\sqrt{5}}{\sqrt{2}}$ $\sqrt{\frac{2}{3}}$

$=\sqrt{\frac{5}{3}}$