Learning Objectives
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
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.
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.
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.
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.
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 .
What do you think it is?
Let's find out if you are right:
Since
cubed is , is
the cube root of .
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 .
What do you think it is?
Let's find out if you are right:
Since
squared is , is
the square root of .
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
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.
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.
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
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?
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 19, 2011 by Kim Seward.
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