## Simplifying Expressions Using the Product Property For Exponents

Contents

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.

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:

\(\begin{array}{c}\hfill {x}^{2}·{x}^{3}\hfill \\ \hfill {x}^{2+3}\hfill \\ \hfill {x}^{5}\hfill \end{array}\)

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

### Product 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 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.

\(\begin{array}{ccc}\hfill {2}^{2}·{2}^{3}& \stackrel{?}{=}\hfill & {2}^{2+3}\hfill \\ \hfill 4·8& \stackrel{?}{=}\hfill & {2}^{5}\hfill \\ \hfill 32& =\hfill & 32\phantom{\rule{0.2em}{0ex}}\text{✓}\hfill \end{array}\)

## Example

Simplify: \({y}^{5}·{y}^{6}.\)

### Solution

Use the product property, a · ^{m}a = ^{n}a.^{m+n} | |

Simplify. |

## Example

Simplify:

- \({2}^{5}·{2}^{9}\)
- \(3·{3}^{4}.\)

### Solution

Use the product property, *a*·^{m}*a*=^{n}*a*.^{m+n}Simplify. Use the product property, *a*·^{m}*a*=^{n}*a*.^{m+n}Simplify.

## Example

Simplify:

- \({a}^{7}·a\)
- \({x}^{27}·{x}^{13}.\)

### Solution

Rewrite, a = a ^{1}.Use the product property, *a*·^{m}*a*=^{n}*a*.^{m+n}Simplify. Notice, the bases are the same, so add the exponents. Simplify.

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

## Example

Simplify: \({d}^{4}·{d}^{5}·{d}^{2}.\)

### Solution

Add the exponents, since bases are the same. | |

Simplify. |