Represent the following mixed infinite decimal periodic fractions as common fractions:
Simplify the following expressions.
$({(\frac{y}{y - x})}^{- 2} - \frac{(x + y)^{2} - 4 x y}{x^{2} - x y}) \frac{x^{4}}{x^{2} y^{2} - y^{4}}$

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Solution

The given equation can be written as

$[({\frac{x - y}{y})}^{2} - \frac{{(x - y)}^{2}}{x (x - y)}] \frac{x^{4}}{y^{2} (x^{2} {- y}^{2})}$ [since $(x + y)^{2} - 4 x y = (x - y)^{2}$ ]

$\Rightarrow [\frac{(x - y) (x^{2} {- x y - y}^{2})}{x y^{2}}] [\frac{x^{4}}{y^{2} (x^{2} - y^{2})}] = \frac{x^{3} (x^{2} - x y - y^{2})}{y^{4} (x + y)}$

$\Rightarrow \frac{x^{3} (x^{2} - x y - y^{2})}{y^{4} (x + y)} = \frac{x^{5} - x^{4} y - x^{2} y^{2}}{x y^{4} + y^{5}}$

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