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College Algebra
Tutorial 4: Radicals


 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Find the principal nth root of an expression.
  2. Find the nth root of an expression raised to the nth power.
  3. Simplify radical expressions.
  4. Multiply radicals that have the same index number.
  5. Divide radicals that have the same index number.
  6. Add and subtract like radicals.
  7. Rationalize one term denominators of rational expressions.
  8. Rationalize two term denominators of rational expressions.




desk Introduction



In this tutorial we will be looking at radicals (or roots).  Basically, the root of an expression is the reverse of raising it to a power.  For example, if you want the square root of an expression, then you want another expression, such that, when you square it, you get what is inside the square root.  This concept carries through to all roots. We will be looking at rewriting, simplifying, adding, subtracting, multiplying, and rationalizing the denominator of radicals.  You never know when your radical simplifying skills may come in handy, so you want to be prepared.

 

 

desk Tutorial



 

square root

If n is even:
 

If n is even, then a and b must be nonnegative for the root to be a real number. 

If n is even and a is negative, then the root is not a real number.

If n is odd:
 

If n is odd, then a and b can be any real number.

 
 
Things to note about radicals in general:
 
 
When looking for the nth radical or nth root, you want the expression that, when you raise it to the nth power, you would get the radicand (what is inside the radical sign).
 

When there is no index number, n, it is understood to be a 2 or square root.  For example:

square root = principal square root of x.



Note that NOT EVERY RADICAL is a square root.  If there is an index number n other than the number 2, then you have a root other than a square root.


 
 
 

notebook Example 1: Evaluate ex 1a or indicate that the root is not a real number.

videoView a video of this example 


 
The thought behind this is that we are looking for the square root of 100. This means that we are looking for a number that when we square it, we get 100. 

What do you think it is?

Let’s find out if you are right:

ex 1b

Since 10 squared is 100, 10 is the square root of 100.

Note that we are only interested in the principal root and since 100 is positive and there is not a sign in front of the radical, our answer is positive 10.  If there had been a negative in front of the radical our answer would have been -10.


 
 
 

notebook Example 2: Evaluate ex 2a or indicate that the root is not a real number.

videoView a video of this example


 
Now we are looking for the negative of the fourth root of 16, which means we are looking for a number that when we raise it to the fourth power we get 16 (then we will take its negative). 

What do you think it is?

Let’s find out if you are right:

ex 2b

Since 2 raised to the fourth power is 16 and we are negating that, our answer is going to be -2.

Note that the negative was on the outside of our even radical.  If the negative had been on the inside of an even radical, then the answer would be no real number.


 
 
 

notebook Example 3: Evaluate ex 3 or indicate that the root is not a real number.

videoView a video of this example


 
Now we are looking for the square root of -100, which means we are looking for a number that when we square it we get -100.

What do you think it is?

Let’s find out if you are right:

Since there is no such real number that when we square it we get -100, the answer is not a real number.


 
 
 
nth root rule

If n is an even positive integer, then even root
 

If n is an odd positive integer, then odd root


 
If a problem does not indicate that a variable is positive, then you need to assume that we are dealing with both positive and negative real numbers and use this rule.

 
 
 

notebook Example 4: Simplify ex 4a.

videoView a video of this example


 
Since it didn’t say that y is positive, we have to assume that it can be either positive or negative.  And since the root number and exponent are equal, then we can use the nth rootrule.

ex 4b

Since the root number and the exponent inside are equal and are the even number 2, we need to put an absolute value around y for our answer. 

The reason for the absolute value is that we do not know if y is positive or negative.  So if we put y as our answer and it was negative, it would not be a true statement. 

For example if y was -5, then -5 squared would be 25 and the square root of 25 is 5, which is not the same as -5.  The only time that you do not need the absolute value on a problem like this is if it stated that the variable is positive.


 
 
 

notebook Example 5: Simplify ex 5a.

videoView a video of this example


 
Since the root number and exponent are equal, then we can use the nth rootrule.

ex 5b

This time our root number and exponent were both the odd number 3.  When an odd numbered root and exponent match then the answer is the base whether it is negative or positive.


 


 

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 (found below) 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 of radicals (found below) 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.


 

A Product of Two Radicals 
With the Same Index Number

radical product


 
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 6: Use the product rule to simplify ex 6a.

videoView a video of this example


 
ex 6b
*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 15 and 15 does not have any factors we can take the cube root of, this is as simplified as it gets.


 
 
 

notebook Example 7: Use the product rule to simplify ex 7a.

videoView a video of this example


 
ex 7b
*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 is inside the radical sign and 24 does not have any factors we can take the fourth root of, this is as simplified as it gets.


 
 
 

notebook Example 8: Use the product rule to simplify ex 8a.

videoView a video of this example


 
Even though 75 is not a perfect square, it does have a factor that we can take the square root of.

Check it out:


 
ex 8b

*Rewrite 75 as (25)(3)

*Use the prod. rule of radicals to rewrite
*The square root of 25 is 5


 
In this example, we are using the product rule of radicals in reverse to help us simplify the square root of 75.  When you simplify a radical, you want to take out as much as possible.  The factor of 75 that we can take the square root of is 25.  We can write 75 as (25)(3) and then use the product rule of radicals to separate the two numbers.  We can take the square root of the 25 which is 5, but we will have to leave the 3 under the square root.

 
 
 

notebook Example 9: Use the product rule to simplify ex 9a.

videoView a video of this example


 
Even though ex 9b  is not a perfect cube, it does have a factor that we can take the cube root of.

Check it out:


 
ex 9c

*Rewrite ex 9bas ex 9e

*Use the prod. rule of radicals to rewrite
*The cube root of ex 9d is ex 9f
 


 
In this example, we are using the product rule of radicals in reverse to help us simplify the cube root of ex 9b.  When you simplify a radical, you want to take out as much as possible.  The factor of ex 9b that we can take the cube root of is ex 9d .  We can write ex 9bas ex 9e and then use the product rule of radicals to separate the two numbers.  We can take the cube root of ex 9d, which is ex 9f, but we will have to leave the rest of it under the cube root.


 

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.
 

radical quotient


 
 
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 10: Use the quotient rule to simplify ex 10a.

videoView a video of this example


 
ex 10b
*Use the  quotient rule of radicals to rewrite
 

*The cube root of -1 is -1 and the cube root of 27 is 3
 


 
 
 

notebook Example 11: Use the quotient rule to simplify ex11a.

videoView a video of this example


 
ex 11b

 

*Use the  quotient rule of radicals to rewrite

*Simplify the fraction

*Use the prod. rule of radicals to rewrite

*The square root of 4 x squared is 2|x|
 


 
Since we cannot take the square root of 10 and 10 does not have any factors that we can take the square root of, this is as simplified as it gets.


 

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:

like radical

like radical


 
 
 
Adding and Subtracting 
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.  Combine the numbers that are in front of the like radicals and write that number in front of the like radical part. 


 
 

notebook Example 12:   Add  radicals.

videoView a video of this example


 

 
Both radicals are as simplified as it gets.

 
Step 2: Combine like radicals.

 
Note how both radicals are the cube root of 2.  These two radicals are like radicals.

 
ex 12b
*Combine like radicals: 3x + 7x = 10x

 
 
 

notebook Example 13:   Subtract ex 13a.

videoView a video of this example


 

 
The 75 in the second 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 second radical:


 
ex 13b

*Rewrite 75 as (25)(3)
*Use Prod. Rule of Radicals

*Square root of 25 is 5

 


 
The first radical is already in its simplest form.

 
Step 2: Combine like radicals.

 
ex 13c
*Combine like radicals: 4 - 30 = -26

 
 
 
Rationalizing the Denominator 
(with one term)

 
When a radical contains an expression that is not a perfect root, for example, the square root of 3 or cube root of 5,  it is called an irrational number.  So, in order to rationalize the denominator, we need to get rid of all radicals that are in the denominator.

 
 
Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.

 
If the radical in the denominator is a square root, then you multiply by a square root that will give you a perfect square under the radical when multiplied by the denominator.   If the radical in the denominator is a cube root, then you multiply by a cube root that will give you a perfect cube under the radical when multiplied by the denominator and so forth...

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. 

Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent.

 

 

Step 3: Simplify the fraction if needed.

 
Be careful.  You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical.  In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.

 
 
 

notebook Example 14:   Rationalize the denominator ex 14a.

videoView a video of this example


 
Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.

 
Since we have a square root in the denominator, then we need to multiply by the square root of an expression that will give us a perfect square under the radical in the denominator.

Square roots are nice to work with in this type of problem because if the radicand is not a perfect square to begin with, we just have to multiply it by itself and then we have a perfect square.

So in this case we can accomplish this by multiplying top and bottom by the square root of 5:


 
ex 14b

*Mult. num. and den. by sq. root of 5
 
 

*Den. now has a perfect square under sq. root
 


 

AND


 
Step 3: Simplify the fraction if needed.

 
ex 14c
*Sq. root of 25 is 5
 

 


 
Be careful when you reduce a fraction like this.  It is real tempting to cancel the 5 which is on the outside of the radical with the 5 which is inside the radical on the last fraction.  You cannot do that unless they are both inside the same radical or both outside the radical.

 
 
 

notebook Example 15:   Rationalize the denominator ex 15a.

videoView a video of this example


 
Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.

 
Since we have a cube root in the denominator, we need to multiply by the cube root of an expression that will give us a perfect cube under the radical in the denominator. 
 

So in this case, we can accomplish this by multiplying top and bottom by the cube root of ex 15b:


 
ex 15c

 

*Mult. num. and den. by cube root of ex 15b
 
 
 

*Den. now has a perfect cube under cube root

 


 

AND


 
Step 3: Simplify the fraction if needed.

 
ex 15d

 

*Cube root of 8 a cube is 2a

 


 
As discussed in example 14, we would not be able to cancel out the 2 with the 20 in our final fraction because the 2 is on the outside of the radical and the 20 is on the inside of the radical. 

Also, we cannot take the cube root of anything under the radical.  So, the answer we have is as simplified as we can get it.


 
 
 
Rationalizing the Denominator 
(with two terms)

 
Above we talked about rationalizing the denominator with one term.  Again, rationalizing the denominator means to get rid of any radicals in the denominator. 

Because we now have two terms, we are going to have to approach it differently than when we had one term, but the goal is still the same.


 
Step 1: Find the conjugate of the denominator.

 
You find the conjugate of a binomial by changing the sign that is between the two terms, but keep the same order of the terms. 

a + b and a - b are conjugates of each other.

 

Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1.

 
Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent.

When you multiply conjugates together you get:

conjugate

 

 

Step 4: Simplify the fraction if needed.

 
Be careful.  You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical.  In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.

 
 
 

notebook Example 16:   Rationalize the denominator ex 16a.

videoView a video of this example


 
Step 1: Find the conjugate of the denominator.

 
In general the conjugate of a + b is a - b and vice versa.

So what would the conjugate of our denominator be?

It looks like the conjugate is ex 16b.


 
Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1.

 
ex 16c

*Mult. num. and den. by conjugate of den.

*Use distributive prop. to multiply the numerators
*In general, product of conjugates is conjugate

 


 

AND


 
Step 4: Simplify the fraction if needed.

 
ex 16d

 

*Square root of 3 squared is 3
 


 
 
 

notebook Example 17:   Rationalize the denominator ex 17a.

videoView a video of this example


 
Step 1: Find the conjugate of the denominator.

 
In general the conjugate of a + b is a - b and vice versa.

So what would the conjugate of our denominator be?

It looks like the conjugate is ex 17b.


 
Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1.

 
ex 17c

*Mult. num. and den. by conjugate of den.

*Use distributive prop. to multiply the numerators
*In general, product of conjugates is conjugate

 


 

AND


 
Step 4: Simplify the fraction if needed.

 
ex 17d
*Square root of 5 squared is 5
*Square root of 7 squared is 7
 
 

*Divide BOTH terms of num. by -2
 

 

 

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 - 1d: Evaluate or indicate that the root is not a real number.
 
 


 
 

 

 

pencilPractice Problems 2a - 2b: Use the product rule to simplify the expression.


 

 

 

pencil Practice Problems 3a - 3b: Use the quotient rule to simplify the expression

 

 

 

pencil Practice Problems 4a - 4b: Add or subtract. 

 

 

 

pencil Practice Problems 5a - 5b: Rationalize the denominator.


 

 

desk Need Extra Help on these Topics?


 
The following are webpages that can assist you in the topics that were covered on this page:
 
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut37_radical.htm
This webpage goes through how to find the nth root of an expression.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut39_simrad.htm
This webpage covers how to multiply, divide, and simplify radical expressions.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut40_addrad.htm
This webpage helps you with adding and subtracting like radicals.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut41_rationalize.htm
This webpage explains how to rationalize denominators.

http://www.purplemath.com/modules/radicals.htm
This website goes over radicals and rationalizing denominators.


 

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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Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 6, 2009 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.