# How do you simplify ((x/y) - (4/x)) / ((1/x) - (2/y^2))?

May 17, 2017

$\frac{y \left({x}^{2} - 4 y\right)}{{y}^{2} - 2 x}$

#### Explanation:

Before you can do the division of the numerator by the denominator, you have to have one fraction at the top and one at the bottom.
The calculation can be shown as:

$\text{numerator " div " denominator}$

$\text{ "(x/y - 4/x) div (1/x - 2/y^2)" } \leftarrow$ find common denominators

$\left(\frac{x}{y} \times \textcolor{red}{\frac{x}{x}} - \frac{4}{x} \times \textcolor{red}{\frac{y}{y}}\right) \div \left(\frac{1}{x} \times \textcolor{red}{{y}^{2} / {y}^{2}} - \frac{2}{y} ^ 2 \times \textcolor{red}{\frac{x}{x}}\right)$

Each fraction has been $\textcolor{red}{\text{multiplied by} 1}$

$\frac{{x}^{2} - 4 y}{x y} \div \frac{{y}^{2} - 2 x}{x {y}^{2}}$

To divide fractions, multiply by the reciprocal of the second fraction:

$\frac{{x}^{2} - 4 y}{x y} \div \frac{x {y}^{2}}{{y}^{2} - 2 x} \text{ } \leftarrow$ now simplify:

$\frac{y \left({x}^{2} - 4 y\right)}{{y}^{2} - 2 x}$