Intermediate Algebra Tutorial 38


Intermediate Algebra
Tutorial 38: Rational Exponents


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deskLearning Objectives


 
After completing this tutorial, you should be able to:
  1. Rewrite a rational exponent in radical notation.
  2. Simplify an expression that contains a rational exponent.
  3. Use rational exponents to simplify a radical expression.




desk 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.

 

 

desk Tutorial


 

 

Rational Exponents and Roots

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

rational exponent
 

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 IIIn 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
 

exponent

exponent

exponent

exponent

exponent

exponent
 
 
 

notebook Example 1: Use radical notation to write the expression and simplify. example 1a

 
example 1b
*Rewrite exponent 1/2 as a square root

 
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.
 
 

notebook Example 2: Use radical notation to write the expression and simplify.example 2a

 
example 2b
*Rewrite exponent 2/3 as a cube root being squared
 

 
 

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.
 
 
 

notebook Example 3: Use radical notation to write the expression and simplify.example 3a

 
example 3b

 

*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.
 
 
 

notebook Example 4: Write with a positive exponent and simplify. example 4a

  example 4b
*Rewrite as recip. of base raised to pos. exp.
 

*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.
 
 

notebook Example 5: Simplify the expression.  Write with positive exponents only.example 5a

 
example 5b

 

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

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

notebook Example 6: Simplify the expression. Write with positive exponents only. example 6a

 
example 6b

 
 

* Multiply like bases, add. exp
 

* Divide like bases, sub. exp
 

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

notebook Example 7: Multiply. example 7a

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

 
example 7b

 
 

* Multiply like bases, add. exp
 

 
 
 
 
 

notebook Example 8:  Factor the common factor from the given expression. example 8a

 
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. 

 
example 8b
*Factor out x to the 1/4
 

 
 
 
 

notebook Example 9:  Use rational exponents to simplify the radical.  Assume that the variable represents a positive number. example 9a

 
example 9b
*Rewrite eighth root of x squared as x to the 2/8 power
 

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

 
desk 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.

 

pencil Practice Problems 1a - 1c: Use radical notation to write the expression and simplify.

 

1a. problem 1a
(answer/discussion to 1a)
1b. problem 1b
(answer/discussion to 1b)

 

1c. problem 1c
(answer/discussion to 1c)
   

 

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

 

2a. problem 2a
(answer/discussion to 2a)

 

 

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

 

3a. problem 3a
(answer/discussion to 3a)
3b. problem 3b
(answer/discussion to 3b)

 

pencil Practice Problem 4a: Multiply.

 

4a. problem 4a
(answer/discussion to 4a)

 

 

 

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

 

5a. problem 5a
(answer/discussion to 5a)

 

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

 

6a. problem 6a
(answer/discussion to 6a)

 

 

 


desk 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.


 


WTAMU > Virtual Math Lab > Intermediate Algebra


Last revised on July 19, 2011 by Kim Seward.
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