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7-3 Multiplication Properties of Exponents
Warm UpWarm Up
Lesson Presentation
California Standards
PreviewPreview
7-3 Multiplication Properties of Exponents
Warm Up
Write each expression using an exponent.1. 2 • 2 • 2
2. x • x • x • x
3.
Write each expression without using an exponent.
4. 43 5. y2 6. m–4
23
4 • 4 • 4 y • y
7-3 Multiplication Properties of Exponents
2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
California Standards
7-3 Multiplication Properties of Exponents
You have seen that exponential expressions are useful when writing very small or very large numbers. To perform operations on these numbers, you can use properties of exponents. You can also use these properties to simplify your answer.
In this lesson, you will learn some properties that will help you simplify exponential expressions containing multiplication.
7-3 Multiplication Properties of Exponents
7-3 Multiplication Properties of Exponents
Products of powers with the same base can be found by writing each power as a repeated multiplication.
am an = (a a … a) (a a … a)
m factors n factors
= a a … a = am+n
m + n factors
7-3 Multiplication Properties of Exponents
7-3 Multiplication Properties of Exponents
Simplify.
Additional Example 1: Finding Products of Powers
A.
Since the powers have the same base, keep the base and add the exponents.
B.
Group powers with the same base together.
Add the exponents of powers with the same base.
7-3 Multiplication Properties of Exponents
Simplify.
Additional Example 1: Finding Products of Powers
C.
D.
n0
1
Group powers with the same base together.
Add the exponents of powers with the same base.
The first two powers have the same base, so add the exponents.
Add the exponents.
Group the first two powers.
7-3 Multiplication Properties of Exponents
A number or variable written without an exponent actually has an exponent of 1.
Remember!
10 = 101
y = y1
7-3 Multiplication Properties of ExponentsCheck It Out! Example 1
a. Simplify.
Since the powers have the same base, keep the base and add the exponents.
b.
Group powers with the same base together.
Add the exponents of powers with the same base.
7-3 Multiplication Properties of ExponentsCheck It Out! Example 1
Simplify.
c.
Group powers with the same base together.
Add.
7-3 Multiplication Properties of ExponentsCheck It Out! Example 1
Simplify.
d.
Group powers with the same base together.
Divide the first group and add the second group.
Multiply.
7-3 Multiplication Properties of ExponentsAdditional Example 2: Astronomy Application
Light from the Sun travels at about miles per second. It takes about 15,000 seconds for the light to reach Neptune. Find the approximate distance from the Sun to Neptune. Write your answer in scientific notation.
distance = rate time Write 15,000 in scientific notation.
Use the Commutative and Associative Properties to group.
Multiply within each group.mi
Neptune is about 2.79 x 109 miles from the Sun.
7-3 Multiplication Properties of ExponentsCheck It Out! Example 2
Light travels at about 1.86 × 105 miles per second. Find the approximate distance that light travels in one hour. Write your answer in scientific notation.
distance = rate time
Write 3,600 in scientific notation.
Multiply within each group.
Use the Commutative and Associative Properties to group.
Light will travel 6.696 × 108 miles in one hour.
7-3 Multiplication Properties of Exponents
To find a power of a power, you can use the meaning of exponents.
n factors
= a a … a a a … a … a a … a = amn
= am am … am
m factors m factors m factors
n groups of m factors
7-3 Multiplication Properties of Exponents
7-3 Multiplication Properties of Exponents
Simplify.
Additional Example 3: Finding Powers of Powers
Use the Power of a Power Property.
Simplify.
1
Use the Power of a Power Property.
Zero multiplied by any number is zero.
Any number raised to the zero power is 1.
7-3 Multiplication Properties of Exponents
Simplify.
Additional Example 3: Finding Powers of Powers
Use the Power of a Power Property.
Simplify the exponent of the first term.
Since the powers have the same base, add the exponents.
Write with a positive exponent.
C.
7-3 Multiplication Properties of ExponentsCheck It Out! Example 3
Simplify.
Use the Power of a Power Property.
Simplify.
1
Use the Power of a Power Property.
Zero multiplied by any number is zero.
Any number raised to the zero power is 1.
7-3 Multiplication Properties of ExponentsCheck It Out! Example 3c
Simplify.
Use the Power of a Power Property.
Simplify the exponents of the two terms.
Since the powers have the same base, add the exponents.
c.
7-3 Multiplication Properties of Exponents
Powers of products can be found by using the meaning of an exponent.
(ab)n = ab ab … ab
n factors
= a a … a b b … b = anbn
n factors n factors
7-3 Multiplication Properties of Exponents
7-3 Multiplication Properties of Exponents
Additional Example 4: Finding Powers of Products
Simplify.
Use the Power of a Product Property.
Simplify.
Use the Power of a Product Property.
Simplify.
A.
B.
7-3 Multiplication Properties of Exponents
Caution!In Example 4A, the negative sign is not part
of the base. –(2y)2 = –1(2y)2
7-3 Multiplication Properties of Exponents
Additional Example 4: Finding Powers of Products
Simplify.
Use the Power of a Product Property.
Use the Power of a Power Property.
Simplify.
C.
7-3 Multiplication Properties of ExponentsCheck It Out! Example 4
Simplify.
Use the Power of a Product Property.
Simplify.
Use the Power of a Product Property.
Use the Power of a Power Property.
Simplify.
7-3 Multiplication Properties of ExponentsCheck It Out! Example 4
Simplify.
Use the Power of a Product Property.
Use the Power of a Power Property.
Simplify.
Write with a positive exponent.
c.
7-3 Multiplication Properties of Exponents
Lesson Quiz: Part I
Simplify.
1. 32 • 34
3.
5.
7.
2.
4.
6.
(x3)2
7-3 Multiplication Properties of Exponents
Lesson Quiz: Part II
8. The islands of Samoa have an approximate area of 2.9 103 square kilometers. The area of Texas is about 2.3 102 times as great as that of the islands. What is the approximate area of Texas? Write your answer in scientific notation.
6.67 × 105 km2