Represent the following mixed infinite decimal periodic fractions as common fractions:
${(6 - 4 {(\frac{5}{16})}^{0})}^{- 2} + {(\frac{2}{3})}^{- 1} - \frac{3}{4}$

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Solution

${(6 - 4 {(\frac{5}{16})}^{0})}^{- 2} + {(\frac{2}{3})}^{- 1} - \frac{3}{4}$
$= (6 - 4 (1))^{- 2}$ $+ \frac{3}{2} - \frac{3}{4}$ ........ $(x^{0} = 1)$
$= \frac{1}{4} + \frac{3}{2} - \frac{3}{4}$ ...... $x^{- 1} = \frac{1}{x^{1}}$
$= \frac{3}{2} - \frac{2}{4}$
$= \frac{3}{2}$ $- \frac{1}{2} = \frac{3 - 1}{2}$ $= \frac{2}{2} = 1$

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