Beginning Algebra
Tutorial 26: Exponents
Learning Objectives
After completing this tutorial, you should be able to:
 Use the definition of exponents.
 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.

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. 
Tutorial
Definition of 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.
Example
1: Evaluate .


*Write the base 1/4 in a
product 3 times
*Multiply 
Example
2: Evaluate . 

*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. 
Example
3: Evaluate . 

*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:
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,

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

*When mult. like bases you add
your exponents 
Example
5: Use the product rule to simplify the
expression . 

*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.
Example
6: Evaluate . 

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

Example
7: Evaluate . 

*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.
Example
8: Use the quotient rule to simplify the
expression . 

*When div. like bases you subtract
your exponents

Example
9: Use the quotient rule to simplify the
expression . 

*When div. like bases you
subtract your exponents

Base Raised to Two Exponents
(Power Rule for Exponents)
Specific Illustration

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

*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:
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,

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

*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:
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,

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

*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. 
Example
13: Simplify . 

*When mult. like bases you add
your exponents
*When div. like bases you
subtract your exponents

Example
14: Simplify . 

*When raising a product to an
exponent, raise
each base of the product to that exponent
*When div. like bases you
subtract your exponents

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. 
Practice
Problems 1a  1e: Simplify.
Need Extra Help on these Topics?
Last revised on August 2, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.

