Intermediate Algebra
Tutorial 39: Simplifying Radical Expressions

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

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.

Tutorial
A Product of Two Radicals
With the Same Index Number

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. 
Example
1: Use the product rule to multiply . 
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. 
Example
2: Use the product rule to multiply . 
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.

This works in the same fashion as the rule for a product of two radicals.
This rule can also work in either direction. 
Example
3: Use the quotient rule to simplify . 
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. 
Example
4: Use the quotient rule to simplify . 
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. 
Example
5: Simplify . 
Even though 200 is not a perfect square, it does have a factor that
we can take the square root of.
Check it out: 
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. 
Example
6: Simplify . 
Even though
is not a perfect cube, it does have a factor that we can take the cube
root of. Check it out: 
In this example, we are using the product rule of radicals in reverse
to help us simplify the cube root of .
When you simplify a radical, you want to take out as much as possible.
The factor of that we can take the cube root of is . We can write as
and then use the product rule of radicals to separate the two numbers.
We can take the cube root of ,
which is 3 ab, but we will have to leave the
rest of it under the cube root. 
Example
7: Use the quotient rule to divide and then simplify . 
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. 
Example
8: Use the quotient rule to divide and then simplify.
Assume that the variables are positive. 
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 is a
perfect fourth, we are able to take the fourth root of the whole radicand,
which leaves us with nothing under the radical sign. 
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
Problem 1a: Use the product rule to multiply.
Practice
Problem 2a: Use the quotient rule to simplify.
Practice
Problems 3a  3b: Simplify. Assume that the variables are positive.
Practice
Problem 4a: Use the quotient rule to divide and the then simplify.
Assume that the variables are positive.
Need Extra Help on these Topics?
Last revised on July 20, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
