Lesson 3

Lesson 3: Adding and Subtracting Radicals

Focus

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Knowing how to work with a radical expression and simplify it correctly is needed in all branches of science, not just physics and astronomy. For example, chemists often combine different elements and substances. They need, however, to exercise caution in their labs. If a chemical reaction occurs too quickly, it may be violently explosive. If it occurs too slowly, the substance may remain as a pollutant in the environment for decades. When substances A and B combine to produce C, a radical equation involving the rate of reaction and concentration of substances is

In this lesson you will enhance your ability to work with radicals, including adding, subtracting, and simplifying them and their components.

This lesson will help you answer the following inquiry questions:

• How do you add radical expressions?
• How do you subtract radical expressions?

Assessment

• Lesson 3 Assignment
• Module 5 Project

All assessment items you encounter need to be placed in your course folder.

Save a copy of the Lesson 3 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

Materials and Equipment

• calculator

Launch

This section presents questions to help you determine if you have the skills and knowledge to complete this lesson successfully.

Are You Ready?

Answer the questions and check your solutions. If you are experiencing difficulty, you may want to use the information in the Refresher section to clarify concepts before moving on to the Discover section.

Simplify the following variable expressions by adding or subtracting. If an expression cannot be simplified, explain why not.

1. 5x3 + 3x3 Answer
   
   
2. 4x3 − 2x4 Answer
   
   
3. x3 + x2 Answer
   
   
4. 5x3 − 3x3 Answer
   
   
5. 3(x − 2)2 + 4(x − 2)2 Answer
   
   
6. 6(n + 4)3 − 3(n + 4)2 Answer

Refresher

Visit Like Terms to find a definition of like terms and how to work with them.

Go back to the Are You Ready? section, and try the questions again. If you are still having difficulty, contact your teacher.

Discover

In earlier math courses, you learned that (6) + (7) = (6 + 7). Is this true for radicals? Does ?

Try This 1

Use a calculator to complete Adding Radicals Chart.

You will need to include brackets as you enter these calculations in your calculator. For example, to enter you will need to include brackets around the radicands. The following keystrokes will be needed for most calculators.

You also know from earlier math courses that 6(2) + 4(2) = (6 + 4)(2). Is this true for radicals? Does ?

Try This 2

Use a calculator to complete Mixed Radicals Chart. You will likely need to include brackets as you enter the expressions in your calculator.

Share 1

Based on your observations from both of the Try This activities, work with a partner to create a rule that applies to the addition of radical expressions.

Place a copy of your work in your course folder.

Explore

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As you saw in the Discover section, adding and subtracting radicals works the same as adding and subtracting like terms. In the case of radicals, the radicands are considered the like terms. Just as you can’t always simplify a variable expression all the way down to one term, you can’t always simplify a radical expression down to one term. We have all heard the expression “you can’t add apples and oranges.” In order to add or subtract radical terms, they have to have the same radicand and index.

Selena and Josh are working together on their homework, and the assignment is to simplify some radical expressions that require adding and subtracting. They use different methods to simplify the expressions. Look at Selena’s and Josh’s Solutions to decide which method you would prefer to use when simplifying radicals.

Connect

Lesson Assignment

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Lesson 3 Summary

Much like you can add and subtract numbers and variable expressions, you can add and subtract radicals. The key is to ensure that you are working with like terms. Once you simplify each term in the expression, you can add or subtract any terms that have the same radicand and index. If there are terms that do not have the same radicand and index, they cannot be combined. Just as you can’t always simplify a variable expression all the way down to one term, you can’t always simplify a radical expression down to one term.

In the next lesson you will learn how to multiply and divide radical expressions.