Intermediate Algebra
Tutorial 37: Radicals

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Learning Objectives
After completing this tutorial, you should be able to:
 Find the principal nth root of an expression.
 Find the nth root of an expression raised to the nth power.

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.

Tutorial
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:
= 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.


Example
1: Simplify . 
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:
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. 
Example
2: Simplify . 
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:
Since 1/2 cubed is 1/8, our answer is going to be 1/2. 
Example
3: Simplify . 
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:
Since 3 raised to the fourth power is 81, our answer is going to
be 3.

Example
4: Simplify. Assume that variables represent positive
real numbers. 
The thought behind this is that we are looking for the square root
of . This means
that we are looking for an expression that, when we square it, we get .
What do you think it is?
Let's find out if you are right:
Since
squared is and
we have a negative in front of the square root, is
the negative square root of .

Example
5: Simplify. Assume that variables represent positive
real numbers. 
The thought behind this is that we are looking for the cube root of .
This means that we are looking for an expression that, when we cube it,
we get .
What do you think it is?
Let's find out if you are right:
Since
cubed is , is
the cube root of .

Example
6: Simplify. Assume that variables represent positive
real numbers. 
The thought behind this is that we are looking for the fourth root
of . This means
that we are looking for an expression that, when we raise it to the fourth
power, it we get .
What do you think it is?
Let's find out if you are right:
Since
raised to the fourth power is , is
the fourth root of .

Example
7: Simplify. Assume that variables represent positive
real numbers. 
The thought behind this is that we are looking for the square root
of
. This means that we are looking for an expression that, when we square
it, we get .
What do you think it is?
Let's find out if you are right:
Since
squared is , is
the square root of .

Example
8: Simplify. Assume that variables represent positive
real numbers. 
The thought behind this is that we are looking for the fifth root of
. This means that we are looking for an expression that, when we raise
it to the fifth power, we get .
What do you think it is?
Let's find out if you are right:
Since raised
to the fifth power is , is
the fifth root of .

rule
If n is an even positive integer, then
If n is an odd positive integer, then

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. 
Example
9: Simplify. 
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 rule.
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. 
Example
10: Simplify. 
Since the root number and exponent are equal, then we can use the rule.
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.

Example
11: Simplify. 
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 rule.
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. 
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  1d: Simplify. Assume that variables represent positive
real numbers.
Practice
Problems 2a  2b: Simplify.
Need Extra Help on these Topics?
Last revised on July 19, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
