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College Algebra
Tutorial 2: Integer Exponents


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deskLearning Objectives


 
After completing this tutorial, you should be able to:
  1. Use the definition of exponents.
  2. Simplify exponential expressions involving multiplying like bases, zero as an exponent, dividing like bases, negative exponents, raising a base to two exponents, raising a product to an exponent and raising a quotient to an exponent. 




desk Introduction



This tutorial covers the basic definition and rules of exponents.  The rules it covers are the product rule, quotient rule, power rule, products to powers rule, quotients to powers rule, as well as the definitions for zero and negative exponents. Exponents are everywhere in algebra and beyond.  Let's see what we can do with exponents.

 

 

desk Tutorial


 

 
Definition of Exponents
(note there are x's in the product) x = base,    n = exponent

 
 
Exponents are another way to write multiplication.

The exponent tells you how many times a base appears in a PRODUCT.
 
 

notebook Example 1: Evaluate .

videoView a video of this example 

*Write the base -3 in a product 4 times
*Multiply 

 
 

notebook Example 2: Evaluate .

videoView a video of this example

*Negate 3 to the fourth
*Put a - in front of 3 written in a product 4 times
*Multiply

 
Hey, this looks a lot like example 1!!!! 

It may look alike, but they ARE NOT exactly the same.  Can you see the difference between the two??  Hopefully, you noticed that in example 1, there was a ( ) around the - and the 3.  In this problem, there is no ( ).  This means the - is NOT part of the base, so it will not get expanded like it did in example 1. 

It is interpreted as finding the negative or opposite of 3 to the fourth power.


 
 
 
 

notebook Example 3: Evaluate .

videoView a video of this example

*Write the base -1/5 in a product 3 times
*Multiply 

 
 
 
Multiplying Like Bases With Exponents
(The Product Rule for Exponents)

Specific Illustration


 
Let’s first start by using the definition of exponents to help you to understand how we get to the law for multiplying like bases with exponents:

Note that 2 + 3 = 5, which is the exponent we ended up with.  We had 2 x’s written in a product plus another 3 x’s written in the product for a total of 5 x’s in the product.  To indicate that we put the 5 in the exponent.
 

Let's put this idea together into a general rule:


 

Multiplying Like Bases With Exponents
(Product Rule for Exponents)

in general,


 
In other words, when you multiply like bases you add your exponents.

The reason is, exponents count how many of your base you have in a product.  So if you are continuing that product, you are adding on to the exponents.
 

notebook Example 4:   Use the product rule to simplify the expression .

videoView a video of this example


*When mult. like bases you add your exponents

 
 
 

notebook Example 5:   Use the product rule to simplify the expression .

videoView a video of this example


*When mult. like bases you add your exponents


 

Zero as an exponent


 
Except for 0, any base raised to the 0 power simplifies to be the number 1.

Note that the exponent doesn’t become 1, but the whole expression simplifies to be the number 1.

 

notebook Example 6:  Evaluate .

videoView a video of this example


*Any expression raised to the 0 power simplifies to be 1

 
 

notebook Example 7:  Evaluate .

videoView a video of this example

Be careful on this example.  Order of operations says to evaluate exponents before doing any multiplication.  This means we need to find x raised to the 0 power first and then multiply it by -15.

 

*x raised to the 0 power is 1
*Multiply

 
 
 
Dividing Like Bases With Exponents
(Quotient Rule for Exponents)

Specific Illustration


 
Let’s first start by using the definition of exponents to help you to understand how we get to the law for dividing like bases with exponents:

Note how 5 - 2 = 3, the final answer’s exponent.  When you multiply you are adding on to your exponent, so it should stand to reason that when you divide like bases you are taking away from your exponent.

Let's put this idea together into a general rule:


 

Dividing Like Bases With Exponents
(Quotient Rule for Exponents)

in general,
 


 
 
In other words, when you divide like bases you subtract their exponents.

Keep in mind that you always take the numerator’s exponent minus your denominator’s exponent, NOT the other way around.
 

notebook Example 8:  Find the quotient .

videoView a video of this example


*When div. like bases you subtract your exponents

 
 

notebook Example 9:  Find the quotient .

videoView a video of this example


*When div. like bases you subtract your exponents

 


 

Negative Exponents
  or 

 
Be careful with negative exponents.  The temptation is to negate the base, which would not be a correct thing to do. Since exponents are another way to write multiplication and the negative is in the exponent, to write it as a positive exponent we do the multiplicative inverse which is to take the reciprocal of the base.
 

notebook Example 10:  Simplify .

videoView a video of this example

*Rewrite with a pos. exp. by taking recip. of base 
 

*Use def. of exponents to evaluate


 
 

notebook Example 11:  Simplify .

videoView a video of this example

*Rewrite with a pos. exp. by taking recip. of base

*Use def. of exponents to evaluate


 
 

Base Raised to Two Exponents
(Power Rule for Exponents)

Specific Illustration


 
 
Let’s first start by using the definition of exponents as well as the law for multiplying like bases to help you to understand how we get to the law for raising a base to two exponents:

Note how 2 times 3 is 6, which is the exponent of the final answer.   We can think of this as 3 groups of 2, which of course would come out to be 6.


 

Base Raised to two Exponents
(Power Rule for Exponents)
in general,


 
In other words, when you raise a base to two exponents, you multiply those exponents together.

Again, you can think of it as n groups of m if it helps you to remember.
 

notebook Example 12:   Simplify .

videoView a video of this example

*When raising a base to two powers you mult. your exponents

 
 

notebook Example 13:   Simplify .

videoView a video of this example


*When raising a base to two powers you mult. your exponents

*Use the definition of neg. exponents to rewrite as the recip. of base

*Use the def. of exponents to evaluate

 


 
 

A Product Raised to an Exponent
(Products to Powers Rule for Exponents)

Specific Illustration


 
Let’s first start by using the definition of exponents to help you to understand how we get to the law for raising a product to an exponent:

Note how both bases of your product ended up being raised by the exponent of 3.


 

A Product Raised to an Exponent
(Products to Powers Rule for Exponents)
in general,


 
In other words, when you have a PRODUCT (not a sum or difference) raised to an exponent, you can simplify by raising each base in the product to that exponent.
 

notebook Example 14:   Simplify .

videoView a video of this example

*When raising a product to an exponent, raise each base of the product to that exponent

 
 
 

notebook Example 15:   Simplify .

videoView a video of this example


*When raising a product to an exponent, raise each base of the product to that exponent

*Mult. exponents when using power rule for exponents


 
 
 

A Quotient Raised to an Exponent
(Quotients to Powers Rule for Exponents)

Specific Illustration


 
Let’s first start by using the definition of exponents to help you to understand how we get to the law for raising a quotient to an exponent:

Since division is really multiplication of the reciprocal, it has the same basic idea as when we raised a product to an exponent.


 

A Quotient Raised to an Exponent
(Quotients to Powers Rule for Exponents)
in general,


 
 
In other words, when you have a QUOTIENT (not a sum or difference) raised to an exponent, you can simplify by raising each base in the numerator and denominator of the quotient to that exponent.
 

notebook Example 16:   Simplify .

videoView a video of this example


 

 

*When raising a quotient to an exponent, raise each base of the quotient to that exponent

*Use def. of exponents to evaluate


 
 
Of course, we all know that life isn’t so cut and dry.  A lot of times you need to use more than one definition or law of exponents to get the job done.  What we did above was to set the foundation to make sure you have a good understanding of the different ideas associated with exponents.  Next we will work through some problems which will intermix these different laws.

 
 
 
Simplifying an Exponential Expression

 
When simplifying an exponential expression,  write it so that each base is written one time with one POSITIVE exponent

In other words, write it in the most condensed form you can making sure that all your exponents are positive.

A lot of times you have to use more than one rule to get the job done.  As long as you use the rule appropriately you should be fine. 


 

notebook Example 17:    Simplify the exponential expression .

videoView a video of this example

*When div. like bases you subtract your exponents: -2 - (-20) = 18
 


 
 

notebook Example 18:     Simplify the exponential expression .

videoView a video of this example

 


 
 

notebook Example 19:    Simplify the exponential expression .

videoView a video of this example

*Rewrite with a pos. exp. by taking recip. of base

 


 
Be careful going into the last line.   Since b doesn't have a negative exponent, we DO NOT take the reciprocal of b.  The other bases each have a negative exponent, so those bases we have to take the reciprocal of.

 

 

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 - 1f: Simplify the exponential expression.

 


 

 

 

 

desk Need Extra Help on these Topics?



 
The following are webpages that can assist you in the topics that were covered on this page:
 
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut23_exppart1.htm
This website gives the definition of and some of the basic rules for exponents. 

http://www.sosmath.com/algebra/logs/log2/log2.html#shortcuts
This webpage gives the definition of exponents.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut24_exppart2.htm
This website helps you with some of the basic rules for exponents.

http://www.purplemath.com/modules/exponent.htm
This webpage gives an overall review of exponents. 

http://www.ltcconline.net/greenl/courses/152A/polyExp/intexp.htm
This webpage goes over the rules of exponents.

http://www.sosmath.com/algebra/logs/log3/log31/log31.html
This website helps you with the product rule for exponents.

http://www.sosmath.com/algebra/logs/log3/log32/log32.html
This website helps you with the quotient rule for exponents.

http://www.sosmath.com/algebra/logs/log3/log33/log33.html
This website helps you with the rule for raising a base to two 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 Feb. 15, 2008 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.