# 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 four factors of  so  means  This format is known as exponential notation.
In the expression  the exponent tells us how many times we use the base  as a factor.
Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

### Simplify 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.
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 of Exponents to more than two factors.

### Simplify Expressions Using the Power Property of 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 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. Look for a pattern.
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.

### Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

### Key Concepts

1. Exponential Notation


This is read 

2. Product Property of Exponents



3. Power Property for Exponents


4. Product to a Power Property for Exponents




### Practice Makes Perfect

Simplify Expressions with Exponents 
In the following exercises, simplify each expression with exponents.
Simplify Expressions Using the Product Property of Exponents 
In the following exercises, simplify each expression using the Product Property of Exponents.
Simplify Expressions Using the Power Property of Exponents 
In the following exercises, simplify each expression using the Power Property of 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 following monomials.

### 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 objectives?
