Section 61
Rules of Exponents

This chapter reviews the
properties of exponents
. If you want a fast reference, all the properties are listed in a table at the end of this page.
PRODUCT OF POWERS PROPERTY
How do you simplify
7
^{
2
}
×
7
^{
6
}
?



If you recall the way
exponents
are defined, you know that this means:
(7
×
7) × (7
×
7
×
7
×
7
×
7
×
7
)
If we remove the parentheses, we have the product of eight 7s, which can be written more simply as:
7
^{
8
}
This suggests a shortcut: all we need to do is add the exponents!
7
^{
2
}
· 7
^{
6
}
= 7
^{
(2 + 6)
}
= 7
^{
8
}
In general, for all real numbers
a
,
b
, and
c
,


a
^{
b
}
·
a
^{
c
}
=
a
^{
(
b
+
c
)
}
^{
}
If you remember only this one and forget the rest, you can use it to figure out most of the other properties.
ZERO EXPONENTS
Many beginning students think it's weird that anything raised to the power of zero is 1. ("It should be 0!") You can use the
product of powers property
to show why this must be true.
7
^{
0
}
· 7
^{
1
}
= 7
^{
(0 + 1)
}
= 7
^{
1
}
We know 7
^{
1
}
= 7. So, this says that 7
^{
0
}
·
7 = 7. What number times 7 equals 7? If we try 0, we have 0
·
7 = 7. No good.
In general, for all real numbers
a
,
a
≠ 0, we have:
a
^{
0
}
= 1
Note that 0
^{
0
}
is undefined. (
Click here to see why.
)
NEGATIVE EXPONENTS
You can use the product of powers property to figure this one out also. Suppose you want to know what 5
^{
–
2
}
is.
5
^{
–
2
}
× 5
^{
2
}
= 5
^{
(
–
2 + 2)
}
= 5
^{
0
}
We know 5
^{
2
}
= 25, and we know 5
^{
0
}
= 1. So, this says that 5
^{
–
2
}
× 25 = 1. What number times 25 equals 1? That would be its multiplicative inverse, 1/25.
In general, for all real numbers
a
and
b
, where
a
≠ 0, we have:
QUOTIENT OF POWERS PROPERTY
When you multiply two powers with the same base, you add the exponents. So when you
divide
two powers with the same base, you
subtract
the exponents. In other words, for all real numbers
a
,
b
, and
c
, where
a
≠ 0,
What you're really doing here is cancelling common factors from the numerator and denominator. Example:
POWER OF A PRODUCT PROPERTY
When you multiply two powers with the same
exponent
, but different bases, things go a little differently.
3
^{
2
}
·
4
^{
2
}
= (3
·
3)
·
(4
·
4)
Because of the
commutative
and
associative properties
of multiplication, we can rewrite this as
3
^{
2
}
·
4
^{
2
}
= (3
·
4)
·
(3
·
4) = 12
^{
2
}
In general, for all real numbers
a
,
b
, and
c
(as long as both
a
and
c
or both
b
and
c
are not zero):
a
^{
c
}
·
b
^{
c
}
= (
ab
)
^{
c
}
POWER OF A QUOTIENT PROPERTY
This is pretty similar to the last one. By canceling common factors, you can see that:
For all real numbers
a
,
b
, and
c
(as long as
b
≠ 0, and
a
and
c
are not both 0):
RATIONAL EXPONENTS
We've covered positive exponents, negative exponents, and zero exponents. But what if you have an exponent which is not an integer? What, for instance, is
9
^{
1/2
}
?
We can fall back again on the product of powers property to find out:
9
^{
1/2
}
×
9
^{
1/2
}
= 9
^{
(1/2 + 1/2)
}
= 9
^{
1
}
We know 9
^{
1
}
= 9. So we need a number which, when multiplied by itself, gives 9. There are two answers... 3 and
–
3.
9
^{
1/2
}
= ±3
So, exponents of 1/2 work a lot like
square roots
. Similarly,
a
^{
1/3
}
is the same as
and in general
There is an important difference here: the square root of 9 is usually defined as
positive
3, whereas 9
^{
1/2
}
means either 3
or
–
3 (in most books, at least).
RECAP
Zero Exponent Property

a
^{
0
}
=
1
, (
a
≠ 0)

Negative Exponent Property


Product of Powers Property


a
^{
b
}
×
a
^{
c
}
=
a
^{
(
b
+
c
)
}
(
a
≠ 0)

Quotient of Powers Property


Power of a Product Property

a
^{
c
}
×
b
^{
c
}
= (
ab
)
^{
c
}
(
a
,
b
≠ 0)

Quotient of a Product Property


Rational Exponent Property

