Represent the following mixed infinite decimal periodic fractions as common fractions:
$(x^{1 / 3} + y^{1 / 3}) (x^{2 / 3} - x^{1 / 3} y^{1 / 3} + y^{2 / 3})$ for $x = 4 \frac{5}{7}$ , $y = 5 \frac{2}{7}$

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Solution

As we seen the given equation carefully and comparing with
$a^{3} + b^{3} = (a + b) (a^{2} + b^{2} - a b)$
then we get,
$a = \frac{1}{3}, b = \frac{1}{3}$
$\Rightarrow a^{3} + b^{3} = x + y$
then we find that our given expression is compressed as
$(x + y)$
and by just putting the value of x and y we get
$4 \frac{5}{7} + 5 \frac{2}{7} = \frac{33}{7} + \frac{37}{7} = \frac{70}{7} = 10$
Hence, our answer is $10$ .

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