chapter 1 section 7. objectives 1 copyright © 2012, 2008, 2004 pearson education, inc. properties...

Chapter 1 Section 7

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Properties of Real Numbers

Use the commutative properties.

Use the associative properties.

Use the identity properties.

Use the inverse properties.

Use the distributive properties.

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Properties of Real Numbers

While the rules for the order of operations say to add from left to right, we may change the order of the terms and group them in any way we choose without affecting the sum.

There are shortcuts that we use in everyday mathematics. Such shortcuts are justified by the basic properties of addition and multiplication, discussed in this section.

In these properties, a, b, and c represent real numbers.

Slide 1.7-3


Objective 1


Slide 1.7-4


The word commute means to go back and forth. Many people commute to work or to school. If you travel from home to work and follow the same route from work to home, you travel the same distance each time.

Commutative Properties

Addition

Multiplication

a b b a ab ba

Slide 1.7-5



Solution:

2 2 _____x

5 ____x x

x

5

Use a commutative property to complete each statement.

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EXAMPLE 1 Using the Commutative Properties



Objective 2

Slide 1.7-7


When we associate one object with another, we think of those objects as being grouped together. We can group the first two together or the last two together and get the same answer.

Associative Properties

Addition

Multiplication

( ) ( )a b c a b c

( ) ( )ab c a bc

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Use an associative property to complete each statement.

Solution:

5 (2 8) ________

10 ( 8) ( 3) ________

( 5 2) 8

10 ( 8) ( 3)

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EXAMPLE 2 Using the Associative Properties


Solution:Commutative

Is an example of the associative property or the commutative property?

(2 4) 6 (4 2) 6

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EXAMPLE 3 Distinguishing Between Properties


Solution:Find the sum.

43 26 17 24 6 (43 17) (26 24) 6

60 50 6

116

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EXAMPLE 4 Using the Commutative and Associative Properties

167 2

2 12 67

2

1 67

67



Objective 3

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If a child wears a costume on Halloween, the child’s appearance is changed, but his or her identity is unchanged. Likewise, the identity of a real number is left unchanged when identity properties are applied.

The number 0 leaves the identity, or value, of any real number unchanged by addition. So 0 is called the identity element for addition, or the additive identity.

Since multiplication by 1 leaves any real number unchanged, 1 is the identity element for multiplication, or the multiplicative identity.

Identity Properties

Addition

Multiplication

0 and 0a a a a 1 and 1a a a a

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Solution:

Use an identity property to complete each statement.

5 ___ 5

1 1___

3 3

0

1

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EXAMPLE 5 Using the Identity Properties


Solution:

Simplify.

36

48

3 5

8 24

6 6

6 8

3 2

4 2

3

4

13 5

8 24

3 3 5

8 3 24

9 5

24 24

4

24

6

4

4

1

6

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EXAMPLE 6 Using the Identity Property to Simplify Expressions



Objective 4

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Each day before you go to work or school, you probably put on your shoes before you leave. Before you go to sleep at night, you probably take them off, and this leads to the same situation that existed before you put them on. These operations from everyday life are examples of inverse operations.

Inverse Properties

Addition

Multiplication

( ) 0 and 0a a a a 1 1

1 and 1 0)a a aa a

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Solution:

Use an inverse property to complete each statement.

___ 6 0

1___ 1

9

6

9

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EXAMPLE 7 Using the Inverse Properties


Simplify.

Solution:

1 13

2 2y

1 13

2 2y

3y

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EXAMPLE 8 Using Properties to Simplify an Expression


Use the distributive property.

Objective 5

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Use the distributive property to rewrite each expression.

4(3 7)

6( )x y z

3 3a b

4 3 4 7 12 28 40

6 ( 6 ) ( 6 )x y z 6 6 6x y z

3( )a b

Solution:

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EXAMPLE 9 Using the Distributive Property


The everyday meaning of the word distribute is “to give out from one to several.”

Look at the value of the following expressions:

, which equals , or 26

, which equals , or 26.

Since both expressions equal 26,

2(5 )8 2(13)

( )5)2 2(8 10 16

2(5 8) 2(5) 2(8).

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Use the distributive properties.


The distributive property says that multiplying a number a by a sum of numbers b + c gives the same result as multiplying a by b and a by c and then adding the two products.

The distributive property is also valid for multiplication over subtraction.

and( )a b c ab ac ( )b c a ba ca

The distributive property can be extended to more than two numbers.

( )a b c d ab ac ad

The distributive property can be used “in reverse.”

( )ac bc a b c Slide 1.7-22

Use the distributive properties. (cont’d)

Distributive Property

( ) and b+ca b c ab ac a ba ca


Solution:

Write the expression without parentheses.

( 5 8)y 5 8y

Slide 1.7-24

EXAMPLE 10 Using the Distributive Property to Remove (Clear) Parentheses

( )x y z x y z

chapter 1 section 7. objectives 1 copyright © 2012, 2008, 2004 pearson education, inc. properties...

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