Intermediate Algebra Tutorial 40


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



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deskLearning Objectives


 
After completing this tutorial, you should be able to:
  1. Add and subtract like radicals.
  2. Multiply radical expressions.




desk 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.

 

 

desk 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:

example 6b

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. 
 
 
 

notebook Example 1:   Add none.

 
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:
 

example 1a
*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.

 
example 1b
*Combine like radicals

 
 
notebook Example 2:   Add or subtract example 2.

 
Step 1: Simplify the radicals.

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

 
example 2a

*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:
 

example 2b

*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:
 

example 2c

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

*Cube root of 27 b cubed is 3b

 
 
 

Step 2: Combine like radicals.

 
example 1c

 

*Combine like radicals
 
 
 
 

notebook Example 3:   Add example 2d.

 
Step 1: Simplify the radicals.

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

 
example 3a
*Use Prod. Rule of Radicals
 

*Fourth root of 16 is 2
 
 

The second radical is already in simplest form.

 
Step 2: Combine like radicals.

 
example 3b

 
 
 

*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.

 
 
 
notebook Example 4:   Multiply and simplify example 3c.

 
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:

 
example 4a

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

 
 
 
notebook Example 5:   Multiply and simplify.  Assume variable is positive. example 4b

 
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:

 
example 5a
*Use Prod. Rule of Radicals
*Square root of a squared is a
*Combine like radicals

 
 
notebook Example 6:   Multiply and simplify example 5b.

 
Step 1: Multiply the radical expression

AND

Step 2: Simplify the radicals.
  

Using distributive property twice we get:

 
example 6a
*Use Prod. Rule of Radicals
*Combine like radicals

 


 
desk 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.

 

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

 

1a. problem 1a
(answer/discussion to 1a)
1b. problem 1b
(answer/discussion to 1b)

 

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

 

2a. problem 2a
(answer/discussion to 2a)

2b. problem 2b
(answer/discussion to 2b)

 

 

desk 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.


 


WTAMU > Virtual Math Lab > Intermediate Algebra


Last revised on July 21, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.