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LESSON
4.5 Rules of Exponents
Notice what happens when you multiply two powers with the same base.
4 factors
3 factors
a 4 • a 3 = (a • a • a • a) • (a • a • a) = a 4 + 3 = a 7
7 factors
This example suggests a rule for multiplying powers with the same base.
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LESSON
4.5 Rules of Exponents
Product of Powers Property
Words To multiply powers with the same base, add their
exponents.
Algebra a m • a n = a m + n
Numbers 4 3 • 4 2 = 4 3 + 2 = 4 5
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LESSON
4.5 Rules of Exponents
EXAMPLE
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Using the Product of Powers Property
Lake Powell Lake Powell, the reservoir behind the Glen Canyon Dam in
Arizona, can hold about 10 12 cubic feet of water when full. There are about
10 27 water molecules in 1 cubic foot of water. About how many water
molecules can the reservoir hold?
SOLUTION
ANSWER
Number of water
molecules in reservoir
=
Cubic feet of
water in reservoir
•
Number of water
molecules in a cubic foot
= 10 12 • 10 27
Substitute values.
= 10 12 + 27
Product of powers property
= 10 39
Add exponents.
Lake Powell can hold about 10 39 molecules of water.
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LESSON
4.5 Rules of Exponents
EXAMPLE
2
Using the Product of Powers Property
3x • 5x 5 = 3 • 5 • x 1 • x 5
Commutative property of multiplication
= 3 • 5 • x1 + 5
Product of powers property
= 3 • 5 • x6
Add exponents.
= 15x 6
Multiply.
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LESSON
4.5 Rules of Exponents
Quotients of Powers There is a related rule you can use for dividing
powers with the same base. The following example suggests this rule.
5 factors
a5
a2
1
=
a•a•a•a•a
a•a•a•a•a
=
a•a
a•a
1
2 factors
1
= a • a • a = a5 – 2 = a3
1
3 factors
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LESSON
4.5 Rules of Exponents
Quotient of Powers Property
Words To divide powers with the same base, subtract
the exponent of the denominator from the exponent of the
numerator.
Algebra
am
m – n, where a ≠ 0
=
a
n
a
Numbers
68
= 68 – 5 = 63
5
6
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LESSON
4.5 Rules of Exponents
EXAMPLE
3
Using Both Properties of Powers
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2
Simplify 3m • m .
6m 3
3m 5 • m 2
3m 5 + 2
= 6m 3
6m 3
Product of powers property
3m 7
=
6m 3
Add exponents.
3m 7 – 3
=
6
Quotient of powers property
3m 4
=
6
Subtract exponents.
m4
=
2
Divide numerator and denominator by 3.
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