# Multiplication and Fractions with Powers

The terms ${a}^{3}$ and $7{a}^{2}$ cannot be simplified. These powers are of different degrees, so they don’t have the same variable combination! Remember—you can’t simplify two powers even if they have the same variable, if they are of different degrees.

With ${a}^{2}$ and $6{a}^{2}$, on the other hand, those can be simplified to $7{a}^{2}$. This is because the variables are the same—$a$—and they’re the same degree, the second power.

Below, you can see how to cancel equal variables out against each other in a fraction.

Example 1

Simplify $\frac{4{x}^{3}\cdot 2x}{x}$

There are many ways to simplify this expression, and now you’ll look at several rules about calculating with powers.

You begin by simplifying the numerator:

 $4{x}^{3}\cdot 2x=4\cdot 2\cdot {x}^{3}\cdot x=8{x}^{4}$

The expression then becomes

 $\frac{8{x}^{4}}{x}=\frac{8\cdot x\cdot x\cdot x\cdot \text{x}}{\text{x}}=8{x}^{3}$

Example 2

Simplify $\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{4x}{a}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{2a}{x}\phantom{\rule{0.17em}{0ex}}}$

You use the flip formula to simplify the expression, and get

$\begin{array}{llll}\hfill \genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{4x}{a}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{2a}{x}\phantom{\rule{0.17em}{0ex}}}& =\frac{4x}{a}÷\frac{2a}{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{4x}{a}\cdot \frac{x}{2a}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{\text{4}{x}^{2}}{\text{2}{a}^{2}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2{x}^{2}}{{a}^{2}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{4x}{a}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{2a}{x}\phantom{\rule{0.17em}{0ex}}}=\frac{4x}{a}÷\frac{2a}{x}=\frac{4x}{a}\cdot \frac{x}{2a}=\frac{\text{4}{x}^{2}}{\text{2}{a}^{2}}=\frac{2{x}^{2}}{{a}^{2}}$