Mathematics » Introducing Polynomials » Using Multiplication Properties of Exponents

# Simplifying Expressions Using the Power Property of Exponents Continued

## Simplifying Expressions Using the Power Property of Exponents Continued

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: $${\left({x}^{2}\right)}^{3}$$ $${x}^{2\cdot 3}$$ $${x}^{6}$$

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

### Definition: Power Property of Exponents

If $$a$$ is a real number and $$m,n$$ 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.

$$\begin{array}{ccc}\hfill {\left({5}^{2}\right)}^{3}& \stackrel{?}{=}& {5}^{2·3}\hfill \\ \hfill {\left(25\right)}^{3}& \stackrel{?}{=}& {5}^{6}\hfill \\ \hfill 15,625& =& 15,625✓\hfill \end{array}$$

## Example

Simplify:

1. $$\phantom{\rule{0.2em}{0ex}}{\left({x}^{5}\right)}^{7}$$
2. $$\phantom{\rule{0.2em}{0ex}}{\left({3}^{6}\right)}^{8}$$

### Solution

 $${\left({x}^{5}\right)}^{7}$$ Use the Power Property, $${\left({a}^{m}\right)}^{n}={a}^{m·n}.$$ Simplify. $${x}^{35}$$
 $${\left({3}^{6}\right)}^{8}$$ Use the Power Property, $${\left({a}^{m}\right)}^{n}={a}^{m·n}.$$ Simplify. $${3}^{48}$$