7-3 Multiplication Properties of Exponents Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

7-3 Multiplication Properties of Exponents

Warm UpWarm Up

Lesson Presentation

California Standards

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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


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


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.



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



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.


Simplify.

Additional Example 1: Finding Products of Powers

C.

D.

n0

1



The first two powers have the same base, so add the exponents.

Add the exponents.

Group the first two powers.


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.




Simplify.

c.


Add.


Simplify.

d.


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.


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.


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



Simplify.

Additional Example 3: Finding Powers of Powers

Use the Power of a Power Property.

Simplify.

1


Zero multiplied by any number is zero.

Any number raised to the zero power is 1.


Simplify.

Additional Example 3: Finding Powers of Powers


Simplify the exponent of the first term.

Since the powers have the same base, add the exponents.

Write with a positive exponent.

C.


Simplify.


Simplify.

1


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.


Simplify the exponents of the two terms.

Since the powers have the same base, add the exponents.

c.


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



Additional Example 4: Finding Powers of Products

Simplify.

Use the Power of a Product Property.

Simplify.


Simplify.

A.

B.


Caution!In Example 4A, the negative sign is not part

of the base. –(2y)2 = –1(2y)2


Additional Example 4: Finding Powers of Products

Simplify.



Simplify.

C.


Simplify.


Simplify.



Simplify.


Simplify.



Simplify.

Write with a positive exponent.

c.


Lesson Quiz: Part I

Simplify.

1. 32 • 34

3.

5.

7.

2.

4.

6.

(x3)2


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

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7-3 Multiplication Properties of Exponents Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview