Lesson 6: Exponents
In this lesson, we will briefly review exponents. This discussion begins with some basic concepts and
terminology that will help you to perform more advanced operations later.
Lesson Objectives
After completing this lesson, you will be able to:
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Explain the basic concepts and terminology of exponents.
Explain how to solve problems with exponents.
Describe the properties of exponents.
Bases and Powers
A value can be described as an entity that has a base. That base can be raised to a particular exponent,
or power. Let’s look at some examples.
32 In this example, 3 is the base and 2 is the power. You say “3 to the power of 2.”
23 In this example, 2 is the base and 3 is the power. You say “2 to the power of 3.”
44 In this example, 4 is the base and 4 is the power. You say “4 to the power of 4.”
The power tells you how many times the base will factor within a multiplication; in other words, the
power tells you the number of times to multiply the base by itself. When you multiply the base by the
power shown, it is called exponential expansion.
Here are a few examples of exponential expansion:
32 = 3 × 3 = 9
In this example, the base 3 is multiplied by itself two times.
23 = 2 × 2 × 2 = 8 In this example, the base 2 is multiplied by itself three times.
44 = 4 × 4 × 4 × 4 = 256 In this example, the base 4 is multiplied by itself four times.
IMPORTANT: Do not make the common error of multiplying the base by the power. The power shows
how many times you multiply the base by itself.
Properties of Exponents
Next, we will review the product, quotient, and power rules of exponents, as well as the rule pertaining
to an exponent with a base of zero.
The product rule of exponents states that when you multiply like bases together, you must add the
exponents of those bases to get the correct result.
Here is an example:
a3 • a4
In this example, a is meant to represent a value. In the problem, each value of a has an exponent. In
order to solve the problem, the exponents are first added together.
a3 • a4 = a3 + 4
Then, the problem can be solved by multiplying a by itself to the power shown (7).
a3 • a4 = a3 + 4 = a7
The quotient rule of exponents says that when dividing like bases, you “bring” the smaller exponent to
the base with the larger exponent and then subtract the exponents. This process can sometimes be
more involved than the word “bring” suggests.
Here is an example of a simple problem:
y3 ÷ y5 = y5 – 3 = y2
It is important to remember that the rules you already learned still apply even though you are working
with exponents. For example, if one of the exponents is negative, use the rules that apply to adding
together values with different signs (positive and negative).
y3 ÷ y–5 = y3 – (–5) = y3 + 5 = y8
The power rule of exponents deals with a base raised to a power inside of a quantity, with that quantity
also raised to a power. In this situation, you multiply the two exponents to solve.
Here is an example:
(a3)5 = a3 ∙ 5 = a15
In this example, the value a is being raised to the third power (a multiplied by itself three times); this
result is then being raised to the fifth power. To solve, you will simply multiply the two exponents (3 and
5). Then you can solve for a. (To solve, you would multiply a by itself 15 times.)
Any base raised to the zero power is going to result in a value of 1, with a single exception. If the base
itself is zero and it is raised to the zero power, the result is undefined.
Here are a few examples:
1440 = 1 In this example, the base 144 is raised to the power of 0. The result is 1.
(–8)0 = 1 In this example, the base –8 is raised to the power of 0. The result is 1.
00 = undefined In this example, the base 0 is raised to the power of 0. The result is undefined.
Figure 1 summarizes the basic properties of exponents and provides the general form to follow and an
example of each.
Basic Properties of Exponents
Property
General Form
Example
Product of Powers Property
Power of a Power Property
am  an  am  n
22  23  223  25
a 
2 
Power of a
Property
Quotient
of
Property
Power of
Property
Zero Property
a
m n
 amn
2 3
 2(2)(3)  26
Product
(ab)m  ambm
(2  3)4  24  34
Powers
am
 am n , where a  0
an
24
 242  22
2
2
3
2
23
 3
4
4
0
2 1
Quotient

m
a
am
 m , where b  0
b
b
0
a  1, where a  0
Figure 1—Properties of Exponents
Now, let’s move on to a review of expressions.
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