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

Mar 12, 2017

$\frac{2 {y}^{2}}{x}$

#### Explanation:

A complex fraction such as $\frac{\frac{a}{b}}{\frac{c}{d}}$ can be written in a more familiar way:

$\frac{\frac{a}{b}}{\frac{c}{d}}$ means the same as $\left(\frac{a}{b}\right) \div \left(\frac{c}{d}\right)$

To divide by a fraction is the same as multiplying by its reciprocal.

$\frac{a}{b} \textcolor{\lim e g r e e n}{\div \frac{c}{d}} = \frac{a}{b} \textcolor{\lim e g r e e n}{\times \frac{d}{c}} = \frac{a d}{b c}$

However if you compare this result to the original fraction you will see that we can simplify a complex fraction immediately:

$\frac{\frac{\textcolor{red}{a}}{\textcolor{b l u e}{b}}}{\frac{\textcolor{b l u e}{c}}{\textcolor{red}{d}}} = \frac{\textcolor{red}{a d}}{\textcolor{b l u e}{b c}}$
Applying this to the complex fraction given gives us:

$\frac{\frac{\textcolor{red}{{y}^{4}}}{\textcolor{b l u e}{{x}^{2}}}}{\frac{\textcolor{b l u e}{x {y}^{2}}}{\textcolor{red}{2 {x}^{2}}}} = \frac{\textcolor{red}{2 {x}^{2} \times {y}^{4}}}{\textcolor{b l u e}{{x}^{2} \times x {y}^{2}}} \text{ }$ which now simplifies to:

$= \frac{2 {y}^{2}}{x}$

Mar 12, 2017

$\frac{2 {y}^{2}}{x}$

#### Explanation:

$\frac{{y}^{4} / {x}^{2}}{\frac{x {y}^{2}}{2 {x}^{2}}}$

:.y^4/x^2 xx (2x^2)/(xy^2

$\therefore {a}^{\textcolor{red}{m}} \cdot {a}^{\textcolor{b l u e}{n}} = {a}^{\textcolor{red}{m} + \textcolor{b l u e}{n}}$

$\therefore \frac{{y}^{\textcolor{red}{4 - 2}} \times 2 {x}^{\textcolor{red}{2 - 2}}}{x} ^ \textcolor{red}{1}$

$\therefore {y}^{\textcolor{red}{4 - 2}} \times 2 {x}^{\textcolor{red}{2 - 2 - 1}}$

:.y^color(red)2 xx 2x^color(red)(color(red)-1

$\therefore {y}^{\textcolor{red}{2}} \times 2 \frac{1}{x} ^ \textcolor{red}{1}$

$\therefore \frac{2 {y}^{2}}{x}$