Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{2^{-2}+2^0}{{\left(\frac{1}{2}\right)}^{-2}-5\cdot (-2{)}^{-2}+{\left(\frac{2}{3}\right)}^{-2}}$

Answer 1

$\frac{2^{-2}+2^{\circ }}{{\left(\frac{1}{2}\right)}^{-2}-5\cdot (-2{)}^{-2}+{\left(\frac{2}{3}\right)}^{-2}}$

$=\frac{{\left(\frac{1}{2}\right)}^2+1}{{\left(4\right)}^2-5.{\left(\frac{1}{-2}\right)}^2+{\left(\frac{3}{2}\right)}^2}$

$=\frac{\frac{5}{4}}{16-\frac{5}{4}+\frac{9}{4}}$

$=\frac{\frac{5}{4}}{16+\frac{4}{4}}$

$=\frac{\frac{5}{4}}{\frac{15}{1}}$

$=\frac{5}{4}\times \frac{1}{15}$ $=\frac{1}{12}$