Mathematics » Introducing Polynomials » Using Multiplication Properties of Exponents

# Simplifying Expressions Using the Product Property of Exponents

## Simplifying Expressions Using the Product Property of Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

First, we will look at an example that leads to the Product Property.

 What does this mean?  How many factors altogether? So, we have Notice that 5 is the sum of the exponents, 2 and 3. We write: $${x}^{2}\cdot {x}^{3}$$ $${x}^{2+3}$$ $${x}^{5}$$

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

### Definition: Product Property of Exponents

If $$a$$ is a real number and $$m,n$$ are counting numbers, then

$${a}^{m}·{a}^{n}={a}^{m+n}$$

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

$$\begin{array}{ccc}\hfill {2}^{2}·{2}^{3}& \stackrel{?}{=}& {2}^{2+3}\hfill \\ \hfill 4·8& \stackrel{?}{=}& {2}^{5}\hfill \\ \hfill 32& =& 32✓\hfill \end{array}$$

## Example

Simplify: $${x}^{5}·{x}^{7}.$$

### Solution

 $${x}^{5}·{x}^{7}$$ Use the product property, $${a}^{m}·{a}^{n}={a}^{m+n}.$$ Simplify. $${x}^{12}$$

## Example

Simplify: $${b}^{4}·b.$$

### Solution

 $${b}^{4}·b$$ Rewrite, $$b={b}^{1}.$$ $${b}^{4}·{b}^{1}$$ Use the product property, $${a}^{m}·{a}^{n}={a}^{m+n}.$$ Simplify. $${b}^{5}$$

## Example

Simplify: $${2}^{7}·{2}^{9}.$$

### Solution

 $${2}^{7}·{2}^{9}$$ Use the product property, $${a}^{m}·{a}^{n}={a}^{m+n}.$$ Simplify. $${2}^{16}$$

## Example

Simplify: $${y}^{17}·{y}^{23}.$$

### Solution

 $${y}^{17}·{y}^{23}$$ Notice, the bases are the same, so add the exponents. Simplify. $${y}^{40}$$

We can extend the Product Property of Exponents to more than two factors.

## Example

Simplify: $${x}^{3}·{x}^{4}·{x}^{2}.$$

### Solution

 $${x}^{3}·{x}^{4}·{x}^{2}$$ Add the exponents, since the bases are the same. Simplify. $${x}^{9}$$