Represent the following mixed infinite decimal periodic fractions as common fractions:
${(\frac{1}{3})}^{- 10} \cdot 27^{- 3} + {0.2}^{- 4} \cdot 25^{- 2} + (64^{- 1 / 9})^{- 3}$

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Solution

$(3)^{10} . ((3)^{3})^{- 3} +$ $(\frac{10}{2})^{4}$ $(\frac{1}{25})^{2}$ + $(((\frac{1}{2})^{9})^{\frac{1}{9}})^{- 3}$ {by using the property of $x^{- 1} = \frac{1}{x}$ } and vice varsa
$(3)^{10} . (3)^{- 9} + \frac{10000}{16} . (\frac{1}{25})^{2} + (2)^{3}$
$(3)^{10 - 9} + 1 + 8$
$3 + 1 + 8 = 12.$

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