1. Skill Check
Lesson Presentation
Lesson Quiz

2. Skill Check
= ? 24
2. 14 – 6 = ? 52
3. 6 • 5 = ? 15
Write the fraction in simplest form. 4. 5.

3. a 4 • a 3 = (a • a • a • a) • (a • a • a) = a 4 + 3 = a 7
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 =
a 7
7 factors
This example suggests a rule for multiplying powers with the same base.

4. 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 5

5. EXAMPLE 1
Using the Product of Powers Property
Lake Powell Lake Powell, the reservoir behind the Glen Canyon Dam in Arizona, can hold about cubic feet of water when full. There are about water molecules in 1 cubic foot of water. About how many water molecules can the reservoir hold?
SOLUTION
Number of water
molecules in reservoir =
Cubic feet of
water in reservoir
molecules in a cubic foot •
= • 10 27
Substitute values. =
Product of powers property
= 10 39
Add exponents.
ANSWER
Lake Powell can hold about molecules of water.

6. 2 x6 • x 9 = x 6 + 9 = x 15 Using the Product of Powers Property
EXAMPLE 2
Using the Product of Powers Property
x6 • x 9 =
x 6 + 9
Product of powers property
= x 15
Add exponents.

7. EXAMPLE 2
Using the Product of Powers Property
= x 11
Add exponents.
x 6 + 5
x6 • x 9 =
Product of powers property
3x • 5x 5 =
3 • 5 • x 1 • x 5
Commutative property of multiplication
= 3 • 5 • x 1 + 5
Product of powers property
= 3 • 5 • x 6
Add exponents.
= 15x 6
Multiply.

8. a • a • a • a • a a • a = a 5 = a 2 a • a • a • a • a a • a =
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
a • a • a • a • a
a • a 1 =
a 5 =
a 2
a • a • a • a • a
a • a =
a • a • a =
a 5 – 2 =
a 3
2 factors
3 factors

9. Quotient of Powers Property
Words
To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.
a m
= a m – n, where a = 0
a n
Algebra
= 6 8 – 5 = 6 3
6 8
6 5
Numbers

10. 3 (0.7)6 – 2 = (0.7) 4 Using the Quotient of Powers Property
EXAMPLE 3
Using the Quotient of Powers Property
(0.7)6 – 2
Quotient of powers property
= (0.7) 4
Subtract exponents.

11. 3 = (0.7)6 – 2 = (0.7) 4 Using the Quotient of Powers Property
EXAMPLE 3
Using the Quotient of Powers Property
= (0.7) 4
Subtract exponents.
= (0.7)6 – 2
Quotient of powers property
Quotient of powers property
Subtract exponents.
Divide numerator and denominator by 2.

12. EXAMPLE 4
Using Both Properties of Powers
Simplify
3m 5 • m 2
6m 3 =
3m 5 • m 2
6m 3
3m 5 + 2
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. =
m 4 2
Divide numerator and denominator by 3.

13. Lesson Quiz
Simplify. 1.
2. 3g2 • 2g4
6g6
b16 3. z6 4.
5. Challenge Find a value of x that makes x + 3 • 5x + 4 = a true statement. Explain how you found your answer.
8; Sample answer: Using the product of powers property, the left side of the equation becomes 52x + 7, and therefore 2x + 7 must equal 23. Solving the equation 2x + 7 = 23 gives x = 8.