Question

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

Answer 1

The given equation can be written as

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

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

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