Represent the following mixed infinite decimal periodic fractions as common fractions:
$(2^{3 / 2} + 27 y^{3 / 5}) : ({(\frac{1}{2})}^{- 1 / 2} + 3 y^{1 / 5})$

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Solution

$(2^{3 / 2} + 27 y^{3 / 5}) : ((\frac{1}{2})^{- 1 / 2} + 3 y^{1 / 5}) = \frac{(2^{1 / 2})^{3} + (3 y^{1 / 5})^{3}}{(2^{1 / 2} + 3 y^{1 / 5})} = 2 + 9 y^{2 / 5} - 2^{1 / 2} .3 y^{1 / 5}$

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