1 Notice what happens when you multiply two powers with the same base. a 4 a 3 = ( a a a a ) ( a a a...
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1 Notice what happens when you multiply two powers with the same base. a 4 • a 3 = (a • a • a • a) • (a • a • a) = 4 factors 3 factors 7 factors This example suggests a rule for multiplying powers with the same base. a 4 + 3 = a 7 Rules of Exponents 4. 5 LESSON
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Transcript of 1 Notice what happens when you multiply two powers with the same base. a 4 a 3 = ( a a a a ) ( a a a...
1
Notice what happens when you multiply two powers with the same base.
a 4 • a 3 = (a • a • a • a) • (a • a • a) =
4 factors 3 factors
7 factors
This example suggests a rule for multiplying powers with the same base.
a 4 + 3 = a 7
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Product of Powers Property
To multiply powers with the same base, add their exponents.Words
Algebra a m • a n = a m + n
Numbers 4 3 • 4 2 = 4 3 + 2 = 4 5
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Using the Product of Powers PropertyEXAMPLE 1
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?
= 10 12 • 10 27
Product of powers property
= 10 39
= 10 12 + 27
ANSWER Lake Powell can hold about 10 39 molecules of water.
Substitute values.
Add exponents.
SOLUTION Number of watermolecules in reservoir =
Cubic feet of water in reservoir
• Number of watermolecules in a cubic foot
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Using the Product of Powers PropertyEXAMPLE 2
= 3 • 5 • x 6
Multiply.= 15x 6
Add exponents.
= 3 • 5 • x 1 + 5 Product of powers property
3 • 5 • x 1 • x 5 Commutative property of multiplication3x • 5x 5 =
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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
2 factors
a • a • a = a 5 – 2 =a 5
=a 2a • a • a • a • a
a • a = a • a • a • a • aa • a
1 1
11
=
3 factors
a 3
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Quotient of Powers Property
To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.
Words
Algebra
Numbers
a m
= a m – n, where a ≠ 0 a n
= 6 8 – 5 = 6 36 8
6 5
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Using Both Properties of PowersEXAMPLE 3
Subtract exponents.
Quotient of powers property
Divide numerator and denominator by 3.
Simplify .3m 5 • m 2
6m 3
Add exponents.
Product of powers property =3m 5 • m 2
6m 33m 5 + 2
6m 3
= 3m 7
6m 3
= 3m 7 – 3
6
=3m 4
6
= m 4
2
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