Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{(0. 6)^{0} - (0.1)^{- 1}}{{(\frac{3}{2^{3}})}^{- 1} \cdot {(\frac{3}{2})}^{3} + {(- \frac{1}{3})}^{- 1}}$

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Solution

$\frac{(0. 6)^{0} - (0.1)^{- 1}}{{(\frac{3}{2^{3}})}^{- 1} \cdot {(\frac{3}{2})}^{3} + {(- \frac{1}{3})}^{- 1}}$

$= \frac{1 - 10}{\frac{8}{3} . \frac{27}{8} - 3}$ ..... $(x^{0} = 1)$

$= \frac{- 9}{9 - 3}$

$= \frac{- 9}{6}$

$= \frac{- 3}{2}$

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