Mathematics » Introducing Polynomials » Using Multiplication Properties of Exponents

# Simplifying Expressions Using the Product to a Power Property

## Simplifying Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

 $${\left(2x\right)}^{3}$$ What does this mean? $$2x·2x·2x$$ We group the like factors together. $$2·2·2·x·x·x$$ How many factors of 2 and of $$x?$$ $${2}^{3}·{x}^{3}$$ Notice that each factor was raised to the power. $${\left(2x\right)}^{3}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}{2}^{3}·{x}^{3}$$ We write: $${\left(2x\right)}^{3}$$ $${2}^{3}·{x}^{3}$$

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

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

If $$a$$ and $$b$$ are real numbers and $$m$$ is a whole number, then

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

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

$$\begin{array}{ccc}\hfill {\left(2·3\right)}^{2}& \stackrel{?}{=}& {2}^{2}·{3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}& 4·9\hfill \\ \hfill 36& =& 36✓\hfill \end{array}$$

## Example

Simplify: $${\left(-11x\right)}^{2}.$$

### Solution

 $${\left(-11x\right)}^{2}$$ Use the Power of a Product Property, $${\left(ab\right)}^{m}={a}^{m}{b}^{m}.$$ Simplify. $$121{x}^{2}$$

## Example

Simplify: $${\left(3xy\right)}^{3}.$$

### Solution

 $${\left(3xy\right)}^{3}$$ Raise each factor to the third power. Simplify. $$27{x}^{3}{y}^{3}$$