Mathematics » Polynomials II » Use Multiplication Properties of Exponents

# Simplifying Expressions Using the Power Property For Exponents

## Simplifying Expressions Using the Power Property For Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

 What does this mean? How many factors altogether? So we have Notice that 6 is the product of the exponents, 2 and 3.

We write:

$$\begin{array}{c}\hfill {\left({x}^{2}\right)}^{3}\hfill \\ \hfill {x}^{2·3}\hfill \\ \hfill {x}^{6}\hfill \end{array}$$

We multiplied the exponents. This leads to the Power Property for Exponents.

### Power Property for Exponents

If $$a$$ is a real number, and $$m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$$ are whole numbers, then

$${\left({a}^{m}\right)}^{n}={a}^{m·n}$$

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

$$\begin{array}{ccc}\hfill {\left({3}^{2}\right)}^{3}& \stackrel{?}{=}\hfill & {3}^{2·3}\hfill \\ \hfill {\left(9\right)}^{3}& \stackrel{?}{=}\hfill & {3}^{6}\hfill \\ \hfill 729& =\hfill & 729\phantom{\rule{0.2em}{0ex}}\text{✓}\hfill \end{array}$$

## Example

Simplify: (a) $${\left({y}^{5}\right)}^{9}$$ (b) $${\left({4}^{4}\right)}^{7}.$$

### Solution

(a)
 Use the power property, (am)n = am·n. Simplify.
(b)
 Use the power property. Simplify.

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