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Intermediate Algebra
Tutorial 39: Simplifying Radical Expressions
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deskLearning Objectives

 
After completing this tutorial, you should be able to:
  1. Multiply radicals that have the same index number.
  2. Divide radicals that have the same index number.
  3. Simplify radical expressions.




desk Introduction


In this tutorial we will be looking at rewriting and simplifying radical expressions.  Part of simplifying radicals is being able to take the root of an expression which is something that is shown in  Tutorial 37: Radicals.  It is good to be comfortable with radicals before entering College Algebra. I think you are ready to move ahead into the tutorial.

 

 

desk Tutorial

 

 

A Product of Two Radicals 
With the Same Index Number

product rule


 
In other words, when you are multiplying two radicals that have the same index number, you can write the product under the same radical with the common index number.

Note that if you have different index numbers, you CANNOT multiply them together.

Also, note that you can use this rule in either direction depending on what your problem is asking you to do.

 
 
notebook Example 1: Use the product rule to multiply example 1a.

 
example 1b
*Use the prod. rule of radicals to rewrite

 

 
Note that both radicals have an index number of 3, so we were able to put their product together under one radical keeping the 3 as its index number. 

Since we cannot take the cube root of 6 and 6 does not have any factors we can take the cube root of, this is as simplified as it gets.

 
 
 
 
notebook Example 2: Use the product rule to multiply example 2a.

 
example 2b
*Use the prod. rule of radicals to rewrite

 

 
Note that both radicals have an index number of 4, so we were able to put their product together under one radical keeping the 4 as its index number. 

Since we cannot take the fourth root of what's inside the radical sign and 10 does not have any factors we can take the fourth root of, this is as simplified as it gets.

 


 

A Quotient of Two Radicals 
With the Same Index Number

If n is even, x and y represent any nonnegative real number
and y does not equal 0.

If n is odd, x and y represent any real number and y does not equal 0.

quotient rule


 
 
This works in the same fashion as the rule for a product of two radicals. 

This rule can also work in either direction.

 
 
 
 
notebook Example 3: Use the quotient rule to simplify example 3a.

 
example 3b
*Use the  quotient rule of radicals to rewrite
 

*Square root of 25 is 5
 

 
Since we cannot take the square root of 2 and 2 does not have any factors that we can take the square root of, this is as simplified as it gets.

 
 
notebook Example 4: Use the quotient rule to simplify example 4a.

 
example 4b
*Use the  quotient rule of radicals to rewrite
 

*The cube root of 8 is 2
 

 
Since we cannot take the cube root of 5 and 5 does not have any factors that we can take the cube root of, this is as simplified as it gets.

 
 
 
Simplifying a Radical Expression

 
When you simplify a radical, you want to take out as much as possible.

We can use the product rule of radicals in reverse to help us simplify the nth root of a number that we cannot take the nth root of as is, but has a factor that we can take the nth root of.  If there is such a factor, we write the radicand as the product of that factor times the appropriate number and proceed. 

We can also use the quotient rule to simplify a fraction that we have under the radical.

Note that the phrase "perfect square" means that you can take the square root of it.  Just as "perfect cube" means we can take the cube root of the number, and so forth.  I will be using that phrase in some of the following examples.

 
 
notebook Example 5: Simplify example 5a.

 
Even though 200 is not a perfect square, it does have a factor that we can take the square root of.

Check it out:

 
example 5b

*Rewrite 200 as (100)(2)

*Use the prod. rule of radicals to rewrite
*The square root of 100 is 10

 
In this example, we are using the product rule of radicals in reverse to help us simplify the square root of 200.  When you simplify a radical, you want to take out as much as possible.  The factor of 200 that we can take the square root of is 100.  We can write 200 as (100)(2) and then use the product rule of radicals to separate the two numbers.  We can take the square root of the 100 which is 10, but we will have to leave the 2 under the square root.

 
 
notebook Example 6: Simplify example 6a.

 
Even though example 6c  is not a perfect cube, it does have a factor that we can take the cube root of.

Check it out:

 
example 6b

*Rewrite example 6cas example 6e

*Use the prod. rule of radicals to rewrite
*The cube root of example 6d is 3ab
 

 
In this example, we are using the product rule of radicals in reverse to help us simplify the cube root of example 6c.  When you simplify a radical, you want to take out as much as possible.  The factor of example 6c that we can take the cube root of is example 6d .  We can write example 6cas example 6e  and then use the product rule of radicals to separate the two numbers.  We can take the cube root of example 6d, which is 3ab, but we will have to leave the rest of it under the cube root.

 
 
 
 
notebook Example 7: Use the quotient rule to divide and then simplify example 7a.

 
example 7b
*Use the  quotient rule of radicals to rewrite

*Simplify fraction
 

 
Note that both radicals have an index number of 2, so we are able to put their quotient together under one radical keeping the 2 as its index number. 

50/5 simplifies to be 10.  Since we cannot take the square root of 10 and 10 does not have any factors we can take the square root of, this is as simplified as it gets.

 
 
 
notebook Example 8: Use the quotient rule to divide and then simplify.  Assume that the variables are positive. example 8a

 
example 8b
 

*Use the  quotient rule of radicals to rewrite

*Simplify fraction
*Take the fourth root
 

 
Note that both radicals have an index number of 4, so we are able to put their quotient together under one radical keeping the 4 as its index number. 

Since example 8cis a perfect fourth, we are able to take the fourth root of the whole radicand, which leaves us with nothing under the radical sign. 

 


 
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 Problem 1a: Use the product rule to multiply.

 

 

 

pencil Practice Problem 2a: Use the quotient rule to simplify.

 

 

 

pencil Practice Problems 3a - 3b: Simplify.  Assume that the variables are positive.

 


 

pencil Practice Problem 4a: Use the quotient rule to divide and the then simplify. 
Assume that the variables are positive.

 

 

  

 

 

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.


 


Last revised on July 20, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.