# Polynomials

## Use Multiplication Properties of Exponents


### Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example,  means to multiply 2 by itself 4 times, so  means .
Let’s review the vocabulary for expressions with exponents.
Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.
Notice the similarities and differences in ⓐ and ⓑ ! Why are the answers different? As we follow the order of operations in part ⓐ the parentheses tell us to raise the  to the 4th power. In part ⓑ we raise just the 5 to the 4th power and then take the opposite.


### Simplify Expressions Using the Product Property for 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.
First, we will look at an example that leads to the Product Property.


We write:
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
An example with numbers helps to verify this property.
We can extend the Product Property for Exponents to more than two factors.

### Simplify 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.

We write:
We multiplied the exponents. This leads to the Power Property for Exponents.
An example with numbers helps to verify this property.

### Simplify 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?
Notice that each factor was raised to the power and  is .
The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.
An example with numbers helps to verify this property:

### Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.
All exponent properties hold true for any real numbers . Right now, we only use whole number exponents.

### Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

### Key Concepts

1. Exponential Notation


2. Properties of Exponents



### Practice Makes Perfect

Simplify Expressions with Exponents
In the following exercises, simplify each expression with exponents.
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression using the Product Property for Exponents.
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression using the Power Property for Exponents.
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression using the Product to a Power Property.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
Multiply Monomials
In the following exercises, multiply the monomials.
Mixed Practice
In the following exercises, simplify each expression.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals?