## 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. Can you find this pattern?

\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(2x\right)}^{3}\hfill \\ \text{What does this mean?}\hfill & & & \phantom{\rule{4em}{0ex}}2x·2x·2x\hfill \\ \text{We group the like factors together.}\hfill & & & \phantom{\rule{4em}{0ex}}2·2·2·x·x·x\hfill \\ \text{How many factors of 2 and of}\phantom{\rule{0.2em}{0ex}}x?\hfill & & & \phantom{\rule{4em}{0ex}}{2}^{3}·{x}^{3}\hfill \end{array}\)

Notice that each factor was raised to the power and \({\left(2x\right)}^{3}\) is \({2}^{3}·{x}^{3}\).

\(\begin{array}{cccc}\text{We write:}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(2x\right)}^{3}\hfill \\ & & & \phantom{\rule{4em}{0ex}}{2}^{3}·{x}^{3}\hfill \end{array}\)

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

### Product to a Power Property for 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:

\(\begin{array}{ccc}\hfill {\left(2·3\right)}^{2}& \stackrel{?}{=}\hfill & {2}^{2}·{3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}\hfill & 4·9\hfill \\ \hfill 36& =\hfill & 36\phantom{\rule{0.2em}{0ex}}\text{✓}\hfill \end{array}\)

## Example

Simplify:

- \({\left(-9d\right)}^{2}\)
- \({\left(3mn\right)}^{3}.\)

### Solution

Use Power of a Product Property, ( *ab*)=^{m}*a*.^{m}b^{m}Simplify. Use Power of a Product Property, ( *ab*)=^{m}*a*.^{m}b^{m}Simplify.