Mathematics » Polynomials II » Divide 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.

$$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\cfrac{x}{y}\right)}^{3}\hfill \\ \text{This means:}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{x}{y}·\cfrac{x}{y}·\cfrac{x}{y}\hfill \\ \text{Multiply the fractions.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{x·x·x}{y·y·y}\hfill \\ \text{Write with exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{{x}^{3}}{{y}^{3}}\hfill \end{array}$$

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

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

$$\begin{array}{cccc}\text{We write:}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left(\cfrac{x}{y}\right)}^{3}\hfill \\ & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{{x}^{3}}{{y}^{3}}\hfill \end{array}$$

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

### Quotient to a Power Property for Exponents

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

$${\left(\cfrac{a}{b}\right)}^{m}=\cfrac{{a}^{m}}{{b}^{m}}$$

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

$$\begin{array}{ccc}\hfill {\left(\cfrac{2}{3}\right)}^{3}& =\hfill & \cfrac{{2}^{3}}{{3}^{3}}\hfill \\ \hfill \cfrac{2}{3}·\cfrac{2}{3}·\cfrac{2}{3}& =\hfill & \cfrac{8}{27}\hfill \\ \hfill \cfrac{8}{27}& =\hfill & \cfrac{8}{27}✓\hfill \end{array}$$

## Example

Simplify:

(a) $${\left(\cfrac{3}{7}\right)}^{2}$$

(b) $${\left(\cfrac{b}{3}\right)}^{4}$$

(c) $${\left(\cfrac{k}{j}\right)}^{3}.$$

### Solution

(a)
 Use the Quotient Property, $${\left(\cfrac{a}{b}\right)}^{m}=\cfrac{{a}^{m}}{{b}^{m}}$$. Simplify.
(b)
 Use the Quotient Property, $${\left(\cfrac{a}{b}\right)}^{m}=\cfrac{{a}^{m}}{{b}^{m}}$$. Simplify.
(c)
 Raise the numerator and denominator to the third power.