**Learning Objectives**

After completing this tutorial, you should be able to:

- Rewrite a rational exponent in radical notation.
- Simplify an expression that contains a rational exponent.
- Use rational exponents to simplify a radical expression.

** Introduction**

In this tutorial we are going to combine two ideas that have been discussed
in earlier tutorials: exponents and radicals. We will look at how
to rewrite, simplify and evaluate these expressions that contain rational
exponents. What it boils down to is if you have a denominator
in your exponent, it** **is your index or root number. So, if
you need to, **review radicals covered
in Tutorial 37: Radicals**. Also, since we are working with
fractional exponents and they follow the exact same rules as integer exponents,
you will need to be familiar with adding, subtracting, and multiplying
them. If fractions get you down you may want to go to **Beginning Algebra Tutorial 3: Fractions**. To review exponents, you can go to **Tutorial
23: Exponents and Scientific Notation Part I** and **Tutorial
24: Exponents and Scientific Notation Part II**. Let's move
onto rational exponents and roots.

** Tutorial**

**If x is positive, p and q are
integers and q is positive,**

I have found it easier to think of it in two parts. Find the root
part first and then take it to the exponential part if possible.
It makes the numbers a lot easier to work with.

Radical exponents follow the exact same exponent rules as discussed
in **Tutorial 23: Exponents and Scientific
Notation, Part I** and **Tutorial
24: Exponents and Scientific Notation, Part II**. **In those
two tutorials we only dealt with integers, but you can extend those rules
to rational exponents. **

Here is a quick review of those exponential rules:

We are looking for the square root of 4 raised to the 1 power, which
is the same as just saying the square root of 4.

If your exponent's numerator is 1, you are basically just looking for the root (the denominator's exponent).

**Our answer is 2** since the square root of 4 is 2.

In this problem we are looking for the cube root of -27 squared.
Again, I think it is easier to do the root part first if possible.
The numbers will be easier to work with.

**The cube root of -27 is -3 and (-3) squared is 9.**

***Rewrite exponent 3/2 as
a square root being cubed**

In this problem we are looking for the square root of 36/49 cubed.
Again, I think it is easier to do the root part first if possible.
The numbers will be easier to work with.

**The square root of 36/49 is 6/7 and 6/7 cubed is 216/343.**

***Rewrite exponent 5/3 as
a cube root raised to 5th power**

In this problem we are looking for the cube root of 1/8 raised to the
fifth power. Again, I think it is easier to do the root part first
if possible. The numbers will be easier to work with.

**The cube root of 1/8 is 1/2 and 1/2 raised to the fifth power is
1/32.**

***Raise a base to two exponents,
mult. exp.**

***Rewrite as recip. of base
raised to pos. exp.**

*** Multiply like bases, add.
exp**

***Rewrite as recip. of base
raised to pos. exp.**

Since this is a binomial times a binomial, we can use **the
FOIL method** as discussed in **Tutorial
26: Multiplying Polynomials**.

*** Multiply like bases, add.
exp**

Basically, we are factoring out a GCF as discussed in **Tutorial
27: The GCF and Factoring by Grouping**. Remember when you
factor out the GCF, you our doing the reverse of the distributive property.

***Simplify exponent**

***Rewrite exponent 1/4 as
a fourth root**

** 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 - 1c:Use radical notation to write the expression and simplify.

Practice Problem 2a:Write with a positive exponent and simplify.

Practice Problems 3a - 3b:Simplify the expression, write with positive exponents only.

Practice Problem 4a:Multiply.

Practice Problem 5a:Factor the common factor from the given expression.

Practice Problem 6a:Use rational exponents to simplify the radical. Assume that the variable represents a positive number.

** 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.

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