### Unit 7: Operations with Monomials

Starting with this unit, you will work with expressions that consist mostly of letters (variables) and do not seem to have very much to do with numbers. You will see how the rules governing operations with these expressions arise from the properties of operations with numbers, particularly distributive property and order of operations. This unit focuses on expressions called monomials. These are expressions that contain only one term (recall from Unit 1 what "term" means). The fact that monomial contains the Greek word mono, which means one, can help you remember this definition. For example, expression *ab* is a monomial, but *a + b* is not, because it contains two terms.

**Completing this unit should take you approximately 7 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 the monomials.

### 7.1: Algebraic Exponential Expressions

You are already familiar with the concepts of exponents and powers from arithmetic. In this subunit, you will review it, emphasizing the odd and even powers of signed (positive or negative) numbers. Then, the concept of exponents will be applied to variables and variable expressions.

Read this article until the section titled "Product of Powers Property." For this subunit on algebraic exponential expressions, complete practice problems 1-13. The definition of a power will be used in the following subunit to derive all of the rules you need to work with exponential expressions. Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. This video points out the difference in exponential expressions such as (-2)4and -24 and explains why the first one equals a positive number while the second one equals a negative.

### 7.2: Rules of Exponents

### 7.2.1: Product of Exponents, Power of Exponent, and Power of a Product

In this subunit, you will learn about these three rules of exponential expressions: to multiply powers of the same base, ADD the exponents; to raise a power to another power, MULTIPLY the exponents; and to raise a product to a power, EACH factor of the product has to be raised to this power.

Read this article, starting with the section titled "Product of Powers Property." Then, complete practice problems 28-43. Watch the "Exponent Properties Involving Products" video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against the answer key.

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

In this subunit, you will learn about the following two rules of exponential expressions: to divide powers of the same base, SUBTRACT the exponents; and to raise a quotient to a power, BOTH the numerator and denominator have to be raised to this power.

Read this page. After reading the article, complete the practice problems 1-26. Once you have completed the practice problems, check your answers against the answer key.

### 7.2.3: Negative Exponents

In this subunit, the definition of a power will be extended to include negative exponents. While this might seem counterintuitive (one cannot multiply a number by itself a negative number of times), there is a way to raise a number to a negative power by taking a reciprocal of the same number raised to a positive power. You will learn why this definition makes sense and learn to apply it to simplify exponential expressions.

Watch this video and take notes. In this video, Sal Khan explains why the reasoning behind the definition of a negative power as a reciprocal of a positive power. This lecture will also help you understand why the rules of exponents that you have learned so far apply to the negative exponents as well.

Read this page. This article provides a lot of practice applying the definition of negative exponents and simplifying expressions containing negative exponents. Using Sample Sets A, B, and C for guidance, complete the exercises in Practice Sets A, B, and C. Then, complete exercises 33-48. You may access the solutions to the problems by clicking on the "Show Solution" link next to the problem.

### 7.3: Simplifying Power Expressions

Watch these videos and take notes. The first video provides a detailed explanation of an example of taking a power of a product. The second video is another example of simplifying exponential expressions, but this time the example involves a quotient. The third example is more complicated, as it involves raising a product to a negative power.

Read the instructions on how to use this page. Solve five problems correctly at the "Middle" level, and then move up to "Difficult" and correctly solve 5 more problems. Use the "Hint" button if you are not sure where to start. The link "Information about Exponent Rules" at the bottom of the page provides a review and examples of all the relevant rules of exponents.

Scroll down to the Exercises section, and complete exercises 133-143. Click on the "Show Solution" button for each problem to check your answer.

Complete this exercise set. This is another opportunity to review and practice simplifying exponential expressions.

### 7.4: Multiplying and Dividing Monomials

Monomials are expressions composed of a product of a number and variables in positive exponents. Multiplying and dividing monomials are the first steps in performing operations with polynomials, which will be discussed in Unit 8.

### 7.4.1: Multiplying Monomials

Watch this video and take notes. In this video, Dr. Sousa explains how to multiply monomials using the rules of exponents.

Read the section titled "Guidance," and work through Example A. Then, complete practice problems 1-5. Once you have completed the practice problems, check your answers against the answer key.

### 7.4.2: Monomial Division

Read these examples and practice problems on monomial division. The answers to the practice problems are provided at the end of the slides.