Transcript 7-3 Multiplication properties of Exponents

7-3 Multiplication Properties of Exponents
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Warm Up
California Standards
Lesson Presentation
7-3 Multiplication Properties of Exponents
Warm Up
Write each expression using an exponent.
1. 2 • 2 • 2 23
2. x • x • x • x
3.
Write each expression without using an
exponent.
4. 43
5. y2
6. m–4
4•4•4
y•y
7-3 Multiplication Properties of Exponents
California
Standards
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.
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
Additional Example 1: Finding Products of Powers
Simplify.
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
Additional Example 1: Finding Products of Powers
Simplify.
C.
Group powers with the same base
together.
Add the exponents of powers with
the same base.
D.
Group the first two powers.
n0
1
The first two powers have the same
base, so add the exponents.
Add the exponents.
7-3 Multiplication Properties of Exponents
Remember!
A number or variable written without an exponent
actually has an exponent of 1.
10 = 101
y = y1
7-3 Multiplication Properties of Exponents
Check It Out! Example 1
Simplify.
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
Check It Out! Example 1
Simplify.
c.
Group powers with the same base
together.
Add.
7-3 Multiplication Properties of Exponents
Check 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 Exponents
Additional 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
mi
group.
Neptune is about 2.79 x 109 miles from the Sun.
7-3 Multiplication Properties of Exponents
Check 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.
Use the Commutative
and Associative
Properties to group.
Multiply within each
group.
8
Light will travel 6.696 × 10 miles in one hour.
7-3 Multiplication Properties of Exponents
To find a power of a power, you can use the meaning
of exponents.
= am  am  …  am
n factors
= a  a … a  a  a … a  … a  a … a = amn
m factors
m factors
m factors
n groups of m factors
7-3 Multiplication Properties of Exponents
7-3 Multiplication Properties of Exponents
Additional Example 3: Finding Powers of Powers
Simplify.
Use the Power of a Power Property.
Simplify.
Use the Power of a Power Property.
Zero multiplied by any number is
zero.
1
Any number raised to the zero
power is 1.
7-3 Multiplication Properties of Exponents
Additional Example 3: Finding Powers of Powers
Simplify.
C.
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.
7-3 Multiplication Properties of Exponents
Check It Out! Example 3
Simplify.
Use the Power of a Power Property.
Simplify.
Use the Power of a Power Property.
Zero multiplied by any number is
zero.
1
Any number raised to the zero
power is 1.
7-3 Multiplication Properties of Exponents
Check It Out! Example 3c
Simplify.
c.
Use the Power of a Power Property.
Simplify the exponents of the two
terms.
Since the powers have the same
base, add the exponents.
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.
A.
Use the Power of a Product Property.
Simplify.
B.
Use the Power of a Product Property.
Simplify.
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.
C.
Use the Power of a Product Property.
Use the Power of a Power Property.
Simplify.
7-3 Multiplication Properties of Exponents
Check 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 Exponents
Check It Out! Example 4
Simplify.
c.
Use the Power of a Product
Property.
Use the Power of a Power
Property.
Simplify.
Write with a positive
exponent.
7-3 Multiplication Properties of Exponents
Lesson Quiz: Part I
Simplify.
1. 32 • 34
2.
3. (x3)2
4.
5.
6.
7.
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