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Beginning Algebra
Tutorial 26: Exponents

 

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, 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 some of the rules of exponents.  The rules it covers are the product rule, quotient rule, power rule, power of a product rule and power of a quotient 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

exponents
(note there are n  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 example 1a.
 

 
example 1b
*Write the base 1/4 in a product 3 times
*Multiply 

 
 
notebook Example 2:   Evaluate example 2a.

 
example 2b

*Write the base -6 in a product 2 times
*Multiply

 
Note how I included the - when I expanded this problem out.  If the - is inside the ( ) of an exponent, then it is included as part of the base.

 
 
notebook Example 3:   Evaluate example 3a.

 
example 3b
*Negate 6 squared
*Put a - in front of 6 written in a product 2 times
*Multiply

 
Hey, this looks a lot like example 2!!!! 

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 2, there was a ( ) around the - and the 6.  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 2. 

It is interpreted as finding the negative or opposite of 6 squared.

 
 
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:

multiplying 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
(The Product Rule for Exponents)

in general,

multiplying exponents

 
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 example 4a.

 
example 4b
*When mult. like bases you add your exponents

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

 
example 5b

*When mult. like bases you add your exponents


 

Zero as an exponent

zero 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 example 6a.

 
example 6b

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

 
 
notebook Example 7:  Evaluate example 7a.

 
Be careful on this example.  The order of operations shown in Tutorial 4: Introduction to Variable Expressions and Equations 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 3.

 
example 7b

*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:

dividing 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,
 

dividing exponents

 
 
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:  Use the quotient rule to simplify the expression example 8a.

 
example 8b

*When div. like bases you subtract your exponents

 
 
notebook Example 9:  Use the quotient rule to simplify the expression example 9a.

 
example 9b
*When div. like bases you subtract your exponents

 

 
 
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:

exponent

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,

exponent

 
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 10:   Use the power rule for exponents to simplify the expression example 10a.

 
example 10b
*When raising a base to 2 powers you mult. your exponents

 
 
A Product Raised to an Exponent
(Power of a Product Rule)

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:

products

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


 

A Product Raised to an Exponent
(Power of a Product Rule)

in general,

products

 
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 11:   Use the power of a product rule to simplify the expression example 11a.

 
example 11b
*When raising a product to an exponent, raise each base of the product to that exponent

 
 
A Quotient Raised to an Exponent
(Power of a Quotient Rule)

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:

quotient

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
(Power of a Quotient Rule)

in general,

quotient

 
 
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 12:   Use the power of a quotient rule to simplify the expression example 12a.

 
example 12b
 

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

*Use def. of exponents to evaluate

 
 
Simplifying an Exponential Expression

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

In other words, write it in the most condense form you can.

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

 
 
notebook Example 13:  Simplify example 13a.

 
 
example 13b
*When mult. like bases you add your exponents
 
 

*When div. like bases you subtract your exponents
 

 

 
 
notebook Example 14:  Simplify example 14a.

 
 
example 14b
*When raising a product to an exponent, raise each base of the product to that exponent
 
 

*When div. like bases you subtract your exponents
 

 

 

 

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 - 1e: Simplify.

 


 

 

 

 

 

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.purplemath.com/modules/exponent.htm
This webpage helps you with the rules of exponents.

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

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

http://www.sosmath.com/algebra/logs/log3/log33/log33.html
This webpage helps with the rule for raising a base to two exponents. 

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


 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

 

 

 

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Last revised on August 2, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.