**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**

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.

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.

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.

***Square root of 25 is 5**

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.

***The cube root of 8 is 2**

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.

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.

Even though 200 is not a perfect square, it does have a factor that
we can take the square root of.

Check it out:

***Use the prod. rule of radicals to rewrite**

***The square root of 100 is 10**

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.

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

Check it out:

***Use the prod. rule of radicals to rewrite**

***The cube root of is **3*ab*

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.

***Simplify fraction**

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.

***Use the quotient
rule of radicals to rewrite**

***Simplify fraction**

***Take the fourth root**

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?**

**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 20, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.