Section 4.6 Negative Exponents
INTRODUCTION
In order to understand negative exponents—the main topic of this section—we need to make sure we
understand the meaning of the reciprocal of a number.
3
Reciprocals of fractions are relatively easy when the fraction is something like 5 :
The reciprocal of
3
5
is
5
3 .
In other words, we simply invert the fraction, “turn it over.”
1
However, the reciprocal of a whole number, such as 7, is the fraction 7 . It’s harder to see the inverted
fraction because with 7 there doesn’t seem to be any fraction at all. Of course, we need to think of 7 as
7
1
1 ; then it’s easier to see its reciprocal: 7 .
Example 1:
a)
Find the reciprocal of each number or variable.
4
9
Answer:
b)
5
-7
c)
6
d)
e)
x
If it isn’t a fraction already, think of it as being over 1, then invert the fraction.
a)
4
9
The reciprocal of 9 is 4
c)
1
reciprocal of 6 is 6 .
Exercise 1
b)
d)
5
The reciprocal of - 7
1
reciprocal of - 8 is - 8 .
7
is - 5
1
reciprocal of x is x .
e)
Find the reciprocal of each number or variable.
a)
2
Reciprocal of 9
is
b)
10
Reciprocal of - 7
d)
Reciprocal of - 5 is
e)
Reciprocal of w is
Negative Exponents
-8
is
c)
Reciprocal of 4 is
f)
Reciprocal of - y is
page 4.6 - 1
UNDERSTANDING NEGATIVE EXPONENTS
Remember this:
Exponents have more meaning than they have value.
In this section we’re going to look at negative exponents, such as x -2, y -5, and 2 -3. However, when the
exponent is negative, such as 2 -3, we can’t really say that there are “negative three factors of 2.” Instead, we
need to find out the meaning of the negative in an exponent, because exponents have more meaning than they
have value.
There are several ways that we can develop the idea of a negative exponent, a few of which will be shown
here. Showing you a variety of methods is not intended to confuse you; instead it is intended to cement the
meaning of the negative exponent, to give it greater validity. Of course, we will be developing a rule, the Rule of
Negative Exponents.
First, let’s start with something rather familiar to us: powers of 3.
We know that
30 = 1
and that
31 = 3
and that multiplying by another 3
3 · 3 = 9 = 32...................
increases the exponent by 1
multiplying by another 3
3 · 3 · 3 = 27 = 33............
increases the exponent by 1
multiplying by another 3
3 · 3 · 3 · 3 = 81 = 34........
increases the exponent by 1
multiplying by another 3
3 · 3 · 3 · 3 · 3 = 243 = 35...
increases the exponent by 1
and so on.
We’ll now consider working in the reverse direction, starting with 35 = 243 and divide by 3, thereby
decreasing the exponent by 1. (Dividing by 3 will mean one less factor of 3.) Of course, dividing by 3 is the
1
1
1
same as multiplying by 3 . So when the process takes us down to 1, we can think of multiplying by 3 : 1 · 3 =
1
1
1
1
1
1
1
3 . Furthermore, we can multiply 3 by 3 and get 9 ; we can then multiply 9 by 3 and get 27 . and on and
on.
Now consider going from one power of 3 to the next by dividing by 3 and let’s take a close look at the
progression of the exponent:
Negative Exponents
page 4.6 - 2
Start with
243 = 35
divide by 3 or multiply by 3
(
1
81 = 34
and decrease the exponent by 1
(
1
)
27 = 33
and decrease the exponent by 1
divide by 3 or multiply by 3
(
1
)
9 = 32
and decrease the exponent by 1
divide by 3 or multiply by 3
(
1
3 = 31
and decrease the exponent by 1
(
1
1 = 30
and decrease the exponent by 1
divide by 3 or multiply by 3
(
1
1
-1
3 = 3
and decrease the exponent by 1
divide by 3 or multiply by 3
(
1
1
-2
9 = 3
and decrease the exponent by 1
(
1
1
-3
27 = 3
and decrease the exponent by 1
(
1
1
-4
81 = 3
and decrease the exponent by 1
)
divide by 3 or multiply by 3
)
)
divide by 3 or multiply by 3
)
)
)
divide by 3 or multiply by 3
)
divide by 3 or multiply by 3
and so on.
Exercise 2
Duplicate the process above using a base of 2; follow the outline given.
32 = 25
Start with
(
1
(
1
(
1
(
1
(
1
(
1
(
1
(
1
)
divide by 2 or multiply by 2
)
divide by 2 or multiply by 2
)
divide by 2 or multiply by 2
)
divide by 2 or multiply by 2
)
divide by 2 or multiply by 2
)
divide by 2 or multiply by 2
)
divide by 2 or multiply by 2
)
divide by 2 or multiply by 2
Negative Exponents
16
=
24
and decrease the exponent by 1
=
and decrease the exponent by 1
=
and decrease the exponent by 1
=
and decrease the exponent by 1
=
and decrease the exponent by 1
=
and decrease the exponent by 1
=
and decrease the exponent by 1
=
and decrease the exponent by 1
page 4.6 - 3
Let’s look graphically at the powers of 2 you just generated. This graph will not be to scale. In fact, the
identity for multiplication, 1, will be treated as the center.
Notice that all of the powers of 2 are to the right of 0. That means that they are all positive.
1
1 1
1
Also notice that all of the values between 0 and 1, such as 16 , 8 , 4 and 2 , are reciprocals.
Please note:
1
- 8, nor does it mean - 8 . The negative in the
exponent has its own meaning, reciprocal.
2 - 3 does not mean
THE RULE OF NEGATIVE EXPONENTS
Here is the Rule of Negative Exponents;
The Rule of Negative Exponents:
1
For any non-zero real number x, x - n = xn .
in other words, xneg. n means the reciprocal of xn .
Example 2:
Rewrite each expression with a positive exponent. Evaluate if possible.
a)
Answer:
Negative Exponents
4 -3
b)
5 -2
c)
7 -1
d)
x -1
Apply the Rule of Negative Exponents. The meaning of the negative in an exponent is
reciprocal.
a)
1
1
4 - 3 = 43 = 64
b)
1
5 - 2 = 52
c)
1
7 - 1 = 71
d)
x -1 =
1
= 7
1
= 25
1
1
x1 = x
page 4.6 - 4
Exercise 3
Rewrite each expression with a positive exponent. Evaluate if possible.
a)
2 -4 =
b)
3 -3 =
c)
7 -2 =
d)
9 -2 =
e)
6 -1 =
f)
10 - 3 =
g)
x -4 =
h)
y -5 =
i)
w-1 =
j)
a -9 =
k)
m-1 =
l)
c -2 =
THE NEGATIVE EXPONENT AND FRACTIONS
At the beginning of this section we were reminded about reciprocals; that is because, as we now know, the
2 negative in the exponent means reciprocal. This is even more evident when the base is a fraction, such as 3
2
2 neg. two
. If we first think of this as 3
, then we can use the meaning of the negative in the exponent to rewrite
3 two
this as its reciprocal raised to a positive power: 2
. We can then evaluate this further using one of the
distributive rules of exponents:
Notice in going from
9
3 2 32
2 = 22 = 4 .
2 neg. two
3 two
3
to 2
, two things happened:
(1) we eliminated the negative in the exponent and
(2) we inverted the fraction
Actually, what happened was, applying the negative exponent to the fraction caused the fraction to invert;
that’s because the negative part of the exponent means reciprocal.
Negative Exponents
page 4.6 - 5
Example 3:
Rewrite each expression with a positive exponent. Evaluate if possible.
a)
Answer:
5
9
-2
b)
4
3
-3
6
5
c)
-1
d)
x
2
-5
Apply the Rule of Negative Exponents. The meaning of the negative in an exponent is
reciprocal. Also apply the one of the rules of distribution.
Exercise 4
5
9
-2
a)
6
5
-1
c)
81
9 2 92
= 5 = 52 = 25
4
3
-3
b)
5
5 1
= 6 = 6
x
2
-5
d)
33
27
3 3
= 4 = 43 = 64
=
32
2 5 25
x = x5 = x5
Rewrite each expression with a positive exponent. Evaluate if possible.
-2
a)
7
4
c)
8
11
e)
6
v
g)
9
5m
=
10
2
1
3
-4
d)
x
4
-2
f)
h)
2x
w
-1
=
-1
=
-2
Negative Exponents
=
-3
b)
=
=
=
-4
=
page 4.6 - 6
ANOTHER WAY TO DEVELOP NEGATIVE EXPONENTS
In the Section 4.1 we saw the quotient rule for exponents:
xa
a – b. This rule is easy to apply for
xb = x
x5
something like x3 = x5 – 3 = x2. It’s also easy to understand when we expand the numerator and denominator
and cancel common factors:
The numerator had five factors of x, but we canceled three of them, leaving it with only two factors of x.
What would happen, though, if the denominator had more factors than the numerator? In other words, what
x3
would happen to something like x5 ? If we were to expand it and cancel, as before, we would get:
Yet, if we were, instead, to use the quotient rule, we’d get
x3
1
So, in one way x5 = x2
and in another way
x3
3 – 5 = x - 2.
x5 = x
x3
1
- 2. Therefore, x - 2 =
=
x
5
x
x2 .
1
xneg. two = xtwo ; the negative means reciprocal and the exponent two
means two factors of x. This time, though, the two factors of x are in the denominator.
Using words and meaning, this says
USING NEGATIVE EXPONENTS WITH THE OTHER RULES.
We can apply the rules learned in Section 4.1 to negative exponents as well as positive exponents. As a
standard, though, it is often request that the end result be written with positive exponents only.
Consider these examples:
Negative Exponents
the product rule:
x5·x - 2 = x5 + (- 2) = x3.
the quotient rule:
x5
5 – (- 2) = x5 + 2 = x7.
x-2 = x
the power rule:
(x 3 )
-4
1
= x3(- 4) = x - 12 = x12
page 4.6 - 7
With negative numbers of any kind, we need to be careful, and that is especially true when working with the
rules of exponents. For the sake of accuracy, it is recommended that you do all of the steps and show all of your
work. Also, think, think, think!
Example 4:
Simplify each expression. Be sure to write the result with positive exponents only.
Answer:
a)
x7·x - 3
b)
w - 2·w - 5
d)
x -3
x -1
e)
(a - 2 )
-3
c)
c3
c -4
f)
(p - 5 )
2
Carefully apply the rules of exponents.
a)
x7·x - 3 = x7 + (- 3) = x4
b)
1
w - 2·w - 5 = w - 2 + (- 5) = w - 7 = w7
c)
c3
c -4
d)
x -3
x -1
e)
(a - 2 )
f)
(p - 5 )
= c3 – (- 4) = c3 + 4 = c7
-3
= a - 2(- 3) = a6
Exercise 5
a)
1
= x - 3 – (- 1) = x - 3 + 1 = x - 2 = x2
2
1
= p - 5(2) = p - 10 = p10
Simplify each expression. Be sure to write the result with positive exponents only.
x9·x - 2
b)
v - 4·v 5
c)
y2·y - 8
d)
w - 3·w - 6
e)
a1
a -6
f)
m-4
m-7
g)
x -6
x -2
h)
y -1
y5
i)
(a - 7 )
j)
(w - 1 )
k)
(p - 2 )
l)
(m 4 )
-2
6
Negative Exponents
-3
-1
page 4.6 - 8
Answers to each Exercise
Section 4.6
Exercise 1:
Exercise 2:
Exercise 4:
Exercise 5:
Negative Exponents
e)
9
2
1
w
i)
8 = 23
ii)
4 = 22
a)
iii)
2 = 21
c)
iv)
1 = 20
e)
v)
1
-1
2 = 2
g)
vi)
1
-2
4 = 2
i)
vii)
1
-3
8 = 2
k)
4
7
2
a)
3
1
4
d)
g)
5m
9
a)
x7
b)
v1 or v
c)
e)
a7
f)
m3
g)
i)
a14
j)
w3
k)
a)
b)
7
- 10
f)
1
-y
1
4
c)
1
-5
d)
Exercise 3:
16
= 49
b)
2
10
= 81
e)
v
6
h)
w
2x
2
=
25m2
81
3
1
1
24
1
72
1
61
1
x4
1
w1
1
m1
8
= 1,000
v
= 6
4
1
= 16
b)
1
= 49
d)
1
= 6
f)
1
1
33 = 27
1
1
92 = 81
1
1
103 = 1,000
1
y5
1
a9
1
c2
h)
1
= w
j)
1
= m
l)
c)
11
8
f)
4
x
2
1
11
= 8
16
= x2
w4
= 16x4
1
y - 6 = y6
1
x - 4 = x4
1
p - 12 = p12
d)
h)
l)
1
w - 9 = w9
1
y - 6 = y6
1
m - 4 = m4
page 4.6 - 9
Section 4.6
1.
Focus Exercises
Rewrite each expression with a positive exponent. (Remember the meaning of the negative exponent.)
Evaluate if possible.
a)
8 -1 =
b)
3 -4 =
c)
11 - 2 =
d)
x -1 =
e)
y -2 =
f)
w-6 =
2.
Rewrite each expression with positive exponents only. (Remember the meaning of the negative exponent.)
Evaluate if possible.
3
5
-1
a)
b
c
-5
d)
3.
-5
=
b)
1
2
=
e)
 1 
 4m 
=
2
 11 
-2
c)
 2x 
w
-4
f)
-3
=
=
=
Simplify each expression. Be sure to write the result with positive exponents only.
a)
m4·m - 3
b)
p - 7·p 6
c)
k9·k - 3
d)
h - 8·h - 5
e)
y1
y -5
f)
m-7
m-4
g)
m-5
m-9
h)
k -1
k6
i)
(x - 1 )
j)
(m 7 )
k)
(h - 1 )
l)
(y - 4 )
-1
Negative Exponents
-8
5
-9
page 4.6 - 10