Intermediate Algebra Tutorial 37


Intermediate Algebra
Tutorial 37: Radicals


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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.




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.  

 

 

desk Tutorial


 
 

nth root
 
 
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: Simplify example 1a.

 
The thought behind this is that we are looking for the square root of 9. This means that, we are looking for a number that when we square it, we get 9. 

What do you think it is?

Let's find out if you are right:

example 1b


Since 3 squared is 9, 3 is the square root of 9.

Note that we are only interested in the principal root and since 9 is positive and there is not a sign in front of the radical, our answer is positive 3.  If there had been a negative in front of the radical, our answer would have been -3.
 
 
 

notebook Example 2: Simplify example 2a.

 
Now we are looking for the third or cube root of -1/8. This means we are looking for a number that, when we cube it, we get -1/8. 

What do you think it is?

Let's find out if you are right:

example 2b


Since -1/2 cubed is -1/8, our answer is going to be -1/2.
 
 

notebook Example 3: Simplify example 3a.

 
Now we are looking for the fourth root of 81. This means that we are looking for a number that, when we raise it to the fourth power,  we get 81. 

What do you think it is?

Let's find out if you are right:

example 3b


Since 3 raised to the fourth power is 81, our answer is going to be 3.
 
 
 

notebook Example 4: Simplify.  Assume that variables represent positive real numbers. example 4a

 
The thought behind this is that we are looking for the square root of example 4b. This means that we are looking for an expression that, when we square it, we get example 4b

What do you think it is?

Let's find out if you are right:

example 4c



Since example 4d  squared is example 4b and we have a negative in front of the square root, -example 4dis the negative square root of example 4b
 
 
 

notebook Example 5: Simplify.  Assume that variables represent positive real numbers. example 5a

 
The thought behind this is that we are looking for the cube root of example 5b. This means that we are looking for an expression that, when we cube it, we get example 5b.

What do you think it is?
 
 

Let's find out if you are right:

example 5c


Since example 5d  cubed is example 5bexample 5dis the cube root of example 5b.  
 
 
 

notebook Example 6: Simplify.  Assume that variables represent positive real numbers. example 6a

 
The thought behind this is that we are looking for the fourth root of example 6b . This means that we are looking for an expression that, when we raise it to the fourth power, it we get example 6b

What do you think it is?

Let's find out if you are right:

example 6c


Since example 6d  raised to the fourth power is example 6bexample 6d  is the fourth root of example 6b.  
 
 
 

notebook Example 7: Simplify.  Assume that variables represent positive real numbers. example 7a

 
The thought behind this is that we are looking for the square root of example 7b    . This means that we are looking for an expression that, when we square it,  we get example 7b

What do you think it is?

Let's find out if you are right:

example 7c



Since example 7d  squared is example 7bexample 7dis the square root of example 7b.  
 
 

notebook Example 8: Simplify.  Assume that variables represent positive real numbers. example 8a

 
The thought behind this is that we are looking for the fifth root of example 8b   . This means that we are looking for an expression that, when we raise it to the fifth power, we get example 8b

What do you think it is?

Let's find out if you are right:

example 8c



Since example 8d raised to the fifth power  is example 8bexample 8d  is the fifth root of example 8b
 
 
  nth root rule

If n is an even positive integer, then nth root
 

If n is an odd positive integer, then nth root
 

The above examples all said that the variables were positive real numbers.  But if a problem does not indicate that, then you need to assume that we are dealing with both positive and negative real numbers and use this rule.

 
 
notebook Example 9: Simplify. example 9a

 
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.

example 9b

Since the root number and the exponent inside are equal and are the even number 2, then 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 as it did on examples 1 - 8 above.
 
 

notebook Example 10: Simplify. example 10a

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

example 10b



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.
 
 
 

notebook Example 11: Simplify. example 11a

 
Since it didn't say that a or b are positive, we have to assume that they can be either positive or negative.  Since the root number and exponent are equal, then we can use the nth rootrule.

example 11b

Since the root number and the exponent inside are equal and are the even number 4, then we need to put an absolute value around a - b for our answer.  The reason for the absolute value is that we do not know if a or b are positive or negative.  So if we put a - b as our answer and it was negative, it would not be a true statement.
 

 
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: Simplify.  Assume that variables represent positive real numbers.

 

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

 

1c. problem 1c
(answer/discussion to 1c)
1d. problem 1d
(answer/discussion to 1d)

 

pencil Practice Problems 2a - 2b: 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.


 


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Last revised on July 19, 2011 by Kim Seward.
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