Learning Objectives
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:
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.
The cube root of -27 is -3 and (-3) squared is 9.
*Rewrite exponent 3/2 as
a square root being cubed
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
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.
* Multiply like bases, add.
exp
*Simplify exponent
*Rewrite exponent 1/4 as
a fourth root
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 - 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.