Mathematics » Introducing Polynomials » Dividing Monomials

# Simplifying Expressions Using the Quotient to a Power Property

## Simplifying Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.

 $${\left(\frac{x}{y}\right)}^{3}$$ This means $$\frac{x}{y}\cdot \frac{x}{y}\cdot \frac{x}{y}$$ Multiply the fractions. $$\frac{x\cdot x\cdot x}{y\cdot y\cdot y}$$ Write with exponents. $$\frac{{x}^{3}}{{y}^{3}}$$

Notice that the exponent applies to both the numerator and the denominator.

We see that $${\left(\frac{x}{y}\right)}^{3}$$ is $$\frac{{x}^{3}}{{y}^{3}}.$$

$$\begin{array}{ccccc}\text{We write:}\hfill & & & & {\left(\frac{x}{y}\right)}^{3}\hfill \\ & & & & \frac{{x}^{3}}{{y}^{3}}\hfill \end{array}$$

This leads to the Quotient to a Power Property for Exponents.

### Definition: Quotient to a Power Property of Exponents

If $$a$$ and $$b$$ are real numbers, $$b\ne 0,$$ and $$m$$ is a counting number, then

$${\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}$$

To raise a fraction to a power, raise the numerator and denominator to that power.

$$\begin{array}{ccc}\hfill {\left(\frac{2}{3}\right)}^{3}& \stackrel{?}{=}& \frac{{2}^{3}}{{3}^{3}}\hfill \\ \hfill \frac{2}{3}\cdot \frac{2}{3}\cdot \frac{2}{3}& \stackrel{?}{=}& \frac{8}{27}\hfill \\ \hfill \frac{8}{27}& =& \frac{8}{27}✓\hfill \end{array}$$

## Example

Simplify:

1. $$\phantom{\rule{0.2em}{0ex}}{\left(\frac{5}{8}\right)}^{2}$$
2. $$\phantom{\rule{0.2em}{0ex}}{\left(\frac{x}{3}\right)}^{4}$$
3. $$\phantom{\rule{0.2em}{0ex}}{\left(\frac{y}{m}\right)}^{3}$$

### Solution

 Use the Quotient to a Power Property, $${\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}$$. Simplify.
 Use the Quotient to a Power Property, $${\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}$$. Simplify.
 Raise the numerator and denominator to the third power.

Did you find this lesson helpful? How can it be improved? Would you like to suggest a correction? Leave Feedback