Intermediate Algebra
Tutorial 40: Adding, Subtracting and Multiplying
Radical Expressions

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Learning Objectives

After completing this tutorial, you should be able to:

Add and subtract like radicals.
Multiply radical expressions.

Introduction

In this tutorial we will look at adding, subtracting and multiplying radical expressions. If you need a review on what radicals are, feel free to go to Tutorial 37: Radicals. If it is simplifying radical expressions that you need a refresher on, go to Tutorial 39: Simplifying Radical Expressions. Ok, I think you are ready to begin this tutorial.

Tutorial

Like Radicals

Like radicals are radicals that have the same root number AND radicand (expression under the root).

The following are two examples of two different pairs of like radicals:

Adding and Subtracting
Radical Expressions

Step 1: Simplify the radicals.

If you need a review on simplifying radicals go to Tutorial 39: Simplifying Radical Expressions.

Step 2: Combine like radicals.

You can only add or subtract radicals together if they are like radicals.

You add or subtract them in the same fashion that you do like terms shown in Tutorial 25: Polynomials and Polynomial Functions. Combine the numbers that are in front of the like radicals and write that number in front of the like radical part.

Example 1: Add

Step 1: Simplify the radicals.

The 20 in the first radical has a factor that we can take the square root of.

Can you think of what that factor is?

Let's see what we get when we simplify the first radical:

*Rewrite 20x as (4)(5x)
*Use Prod. Rule of Radicals
*Square root of 4 is 2

The second radical is already in simplest form.

Step 2: Combine like radicals.

*Combine like radicals

Example 2: Add or subtract

Step 1: Simplify the radicals.

We can take the cube root of b cubed in the first radical:

*Cube root of b cubed is b

The 24 in the second radical has a factor that we can take the cube root of.

Can you think of what that factor is?

Let's see what we get when we simplify the second radical:

*Rewrite 24 as (8)(3)
*Use Prod. Rule of Radicals

*Cube root of 8 is 2

We can take the cube root of the b cubed in the third radical and 81 has a factor that we can take the cube root of.

Can you think of what that factor is?

Let's see what we get when we simplify the third radical:

*Rewrite 81 as (27)(3)
*Use Prod. Rule of Radicals

*Cube root of 27 b cubed is 3b

Step 2: Combine like radicals.

*Combine like radicals

Example 3: Add

Step 1: Simplify the radicals.

We can take the fourth root of the 16 in the first radical:

*Use Prod. Rule of Radicals

*Fourth root of 16 is 2

The second radical is already in simplest form.

Step 2: Combine like radicals.

*Combine like radicals

Multiplying Radical Expressions

Step 1: Multiply the radical expression.

Follow the multiplication property of radicals found in Tutorial 39: Simplifying Radical Expressions and the same basic properties used to multiply polynomials together found in Tutorial 26: Multiplying Polynomials to multiply radical expressions together.

Step 2: Simplify the radicals.

Example 4: Multiply and simplify

Step 1: Multiply the radical expression

AND

Step 2: Simplify the radicals.

Using the distributive property found in Tutorial 5: Properties of Real Numbers we get:

*Use Prod. Rule of Radicals
*Square root of 16 is 4

Example 5: Multiply and simplify. Assume variable is positive.

Step 1: Multiply the radical expression

AND

Step 2: Simplify the radicals.

We can apply the FOIL method found in Tutorial 26: Multiplying Polynomials to this example:

*Use Prod. Rule of Radicals
*Square root of a squared is a
*Combine like radicals

Example 6: Multiply and simplify

Step 1: Multiply the radical expression

AND

Step 2: Simplify the radicals.

Using distributive property twice we get:

*Use Prod. Rule of Radicals
*Combine like radicals

Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problems 1a - 1b: Add or subtract.

1a.

(answer/discussion to 1a)

1b.

(answer/discussion to 1b)

Practice Problems 2a - 2b: Multiply and simplify.

2a.
(answer/discussion to 2a)

2b.
(answer/discussion to 2b)

Need Extra Help on these Topics?

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

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