Intermediate Algebra
Tutorial 38: Rational Exponents

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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
Tutorial
Rational Exponents and Roots
If x is positive, p and q are
integers and q is positive,

In other words, when you have a rational exponent, the denominator
of that exponent is your index or root number and the numerator of the
exponent is the exponential part.
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: 
Review of Exponential Rules

Example
1: Use radical notation to write the expression and simplify. 
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. 
Example
2: Use radical notation to write the expression and simplify. 
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. 
Example
3: Use radical notation to write the expression and simplify. 
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. 
Example
4: Write with a positive exponent and simplify. 
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. 
Example
5: Simplify the expression. Write with positive exponents
only. 
Example
6: Simplify the expression. Write with positive exponents only. 
Example
7: Multiply. 
Example
8: Factor the common factor from the given expression. 
Example
9: Use rational exponents to simplify the radical.
Assume that the variable represents a positive number. 
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?
Last revised on July 19, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
