Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{2^{-2}\cdot 5^3\cdot {10}^{-4}}{2^{-3}\cdot 5^2\cdot {10}^{-5}}$

Answer 1

$\frac{2^{-2}{.5}^3.(2\times 5{)}^{-4}}{2^{-3}{.5}^2.(2\times 5{)}^{-5}}$

$=\frac{(2^{-2}.(2^{-4}).(5{)}^3.(5{)}^{-4}}{\left(2^{-3}\right).\left(2^{-5}\right).(5{)}^2.(5{)}^{-5}}$

$=\frac{2^{-6}{.5}^{-1}}{2^{-8}{.5}^{-3}}$

$=2^{-6+8}{.5}^{-1+3}$

$=2^2{.5}^2$ $=4\times 25=100$