### Unit 7: Operations with Monomials

As we have seen, algebra involves the use of variables to represent unknown quantities in equations. Here we will begin the study of expressions that primarily consist of variables.

The rules that govern operations with these expressions arise from the properties of operations with numbers, such as the distributive property and the order of operations. In this unit we focus on monomials, which are expressions that contain only one term. We will learn how to multiply, divide, and apply rules of exponents to monomials.

**Completing this unit should take you approximately 3 hours.**

Upon successful completion of this unit, you will be able to:

- apply the rules of exponents to simplify algebraic exponential expressions; and
- multiply, divide, and simplify the powers of monomials.

### 7.1: Algebraic Exponential Expressions

The first mathematical functions we will explore are the exponential functions of monomials.

Read this article through the section on the product of powers property. This page reviews writing and solving exponential expressions. Pay attention to how we multiply monomials with exponents, the products of powers property, and example 2.

After you read, complete questions 2, 3, 6, 7, 8, 25, 26, 27 and 28 in the practice set and check your answers.

### 7.2: Manipulating Exponents

In this section, we will look at important rules for manipulating exponents.

Read this article, which shows the rules for multiplying exponents, taking the power of an exponent, and taking the exponent of a product. It may help to write a list of these properties to keep track of them. Watch the video to see a few examples.

After you read, complete questions 2, 3, 6, 7, 14, 15, 25, 26, 33, and 34 in the practice set and check your answers.

### 7.3: Quotient of Exponents and Power of a Quotient

The next important operation is the quotient of exponents, which is what we use when dividing monomials with exponents.

Read this article and pay attention to the quotient of powers property and the power of a quotient property in examples 1 and 2.

After you read, complete questions 1, 2, 5, 6, 12, and 13 in the practice set and check your answers.

### 7.4: Negative Exponents

In this section, we learn how to use negative exponents and how to translate between negative exponents and fractions.

Watch these videos, which walk you through the logic of why negative exponents can be written as fractions.

After you watch, complete this assessment and check your answers.

### 7.5: Multiplying Monomials

Now that you are comfortable with the rules of exponents, you are ready to apply them to functions of monomials. In this section, we will discuss multiplying monomials.

Read the section on multiplying monomials and the solution for example 6.26.

After you read, complete example 6.27 and check your answer.

### 7.6: Dividing Monomials

The last function we must apply to monomials is division.

This article gives an excellent review of the properties of monomials and exponents, as well as how to divide monomials. Review the summary of exponent properties, which goes over everything we discussed in this unit.

After you read, complete examples 6.72 through 6.75 and check your work.