Exponent Properties
Before we look at the exponent properties, we will briefly review some exponent
facts. Exponents represent repeated multiplication. The exponent indicates how
many factors of the base there are. Thus, we can write repeated multiplication in
exponential form. We can also evaluate a value written in exponential form by
writing the repeated factors of the base, and then performing the multiplication.
Example 1: Write the following product in exponential form: 2·2·2
Since there are 3 factors of 2,
2·2·2=23
Example 2: Evaluate 34
Since the exponent is 4, there are 4 factors of 3
34 = 3·3·3·3 = 81
Example 3: Evaluate 71
Since the exponent is 1, there is 1 factor of 7
71 = 7
Using the fact that exponents represent repeated addition, we can simplify the
following:
a3a2 = a3 · a2 = (a·a·a) · (a·a) = a·a·a·a·a = a5
A quicker, more efficient method to arrive at the answer would have been to just add
the exponents: a3a2 = a3+2 = a5
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Product Rule of Exponents: aman = am+n
The Product Rule gives us the total number of factors of the base. It is important to
note that the bases have to be the same.
Example 6: Simplify: x4 · x2 · x
x4 · x2 · x1
x4+2+1
x7
If there no exponent, we insert an exponent of one
The bases are the same, so add the exponents
Example 7: Simplify: (2x3y5z) (5xy2z3)
(2·5) (x3·x1)(y5·y2)(z1·z3)
10 (x3+1)(y5+2)(z1+3)
10x4y7z4
Use the Associative Property
Multiply 2·5 and add the exponents
Again using the fact that exponents represent repeated addition, we will simplify the
following:
(a3)2 = a3 · a3 = a3+3 = a6
A quicker, more efficient method to arrive at the answer would have been to just
multiply the exponents: (a3)2 = a3·2 = a6
Power Rule of Exponents: (am)n = am·n
This can be extended to include the following:
(ab)m = ambm
𝐚 𝐦
(𝐛)
=
𝐚𝐦
𝐛𝐦
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
It is important to be careful to only use the power rule with multiplication inside the
parentheses. This property does not work if there is addition or subtraction.
Basically, in both of the above, we are distributing the outside exponent to the
exponents inside the parentheses.
Example 8: Simplify: (2x5)3
21·3 x5·3
23x15
8x15
Multiply each exponent by 3
Simplify 23
Example 9: Simplify: (-3a4bc2)2
(-3)1·2 a4·2 b1·2 c2·2
(-3)2 a8 b2 c4
9 a8 b2 c4
Example 10: Simplify:
a1·2
b8·2
a2
Multiply each exponent by 2
Simplify (-3)2
a 2
( b8 )
Multiply each exponent by 2
b16
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
4 x2
(y3 z4)
Example 11: Simplify:
41·3 x2·3
3
Multiply each exponent by 3
y3·3 z4·3
4 3 x6
y9 z12
64x6
Simplify 43
y9 z12
Sometimes we will encounter negative exponents. A negative exponent does not
indicate a negative number. Negative exponents yield the reciprocal of the base.
Once we take the reciprocal, the exponent is now positive.
Negative Exponent Rules:
a
-m
=
𝟏
𝟏
𝐚𝐦
𝐚−𝐦
=a
m
𝐚 −𝐦
(𝐛)
=
𝐛 𝐦
(𝐚)
=
𝐛𝐦
𝐚𝐦
As a general rule, if we think of our expression as a fraction, negative exponents in
the numerator must be moved to the denominator, likewise, negative exponents in
the denominator must be moved to the numerator. Once moved, the negative
exponent becomes positive.
Example 12: Simplify: x- 5
x−5
1
1
x5
Place a 1 in the denominator to make a fraction
Move the negative exponent term to the denominator
The negative exponent becomes positive
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 13: Simplify:
33
1
1
3−3
Move the negative exponent term to the numerator
The negative exponent becomes positive
33
33 = 3 ∙ 3 ∙ 3 = 27
Example 14: Simplify: 6y- 8
6y−8
Place a 1 in the denominator to make a fraction
1
Move the negative exponent term to the denominator
The negative exponent becomes positive
6
𝑦8
Consider the following:
a3
a3
a3
a3
= a3 ÷ a3 = 1
any number divided by itself equals 1
= a3 – 3 = a0
using the quotient rule
We can, therefore, conclude that any base to a power of zero is 1.
Zero Power Rule of Exponents: a0 = 1
Example 15: Simplify: 40
40 = 1
Example 16: Simplify: (3x2)0
(3x2)0 = 1
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)