Learning Objectives
Introduction
So, if you need to, review radicals covered in Tutorial 4: 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 you feel that you need a review, click on review of fractions. To review exponents, you can go to Tutorial 2: Integer Exponents. 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 2: Integer Exponents. In that tutorial we only dealt with integers, but you can extend those rules to rational exponents.
Here is a quick review of those exponential rules:
Example 1: Evaluate .
If your exponent's numerator is 1, you are basically just looking for the root (the denominator's exponent).
Our answer is 7 since the square root of 49 is 7.
Example 2: Evaluate .
*Cube root of -125 = -5
The cube root of -125 is -5 and (-5) squared is 25.
Example 3: Evaluate .
*Rewrite as recip. of base
raised to pos. exp.
*DO NOT take the reciprocal of the exponent,
only the base
*Rewrite exponent 3/2 as
a square root being cubed
*Square root of 49/36 = 7/6
From there we are looking for the square root of 49/36 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 49/36 is 7/6 and 7/6 cubed is 343/216.
Example 4: Simplify .
Example 5: Simplify .
*Raise a base to two exponents,
mult. exp.
*Rewrite as recip. of base raised to pos. exp.
*Cube root of 8 = 2
Example 6: Simplify .
Example 7: Simplify by reducing the index of the radical. x represents positive real numbers.
*Simplify exponent
*Rewrite exponent 1/5 as
a fifth 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 - 1b: Evaluate the expression.
Practice Problems 2a - 2c: Simplify the expression.
Practice Problem 3a: Simplify the expression by reducing the index of the radical. x represents positive real numbers.
Need Extra Help on these Topics?
http://www.purplemath.com/modules/exponent5.htm
This webpage assists you with rational exponents.
Go to Get Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for some
more suggestions.
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 6, 2009 by Kim Seward.
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