1.7

Properties of Real Numbers

Use the Commutative Properties

The word commute means to go back and forth. Many people commute to work or to school.

Addition

Multiplication

a b b a

ab ba

The commutative properties say that if two numbers are

added or multiplied in any order, the result is the same.

Pretty Basic

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−8 + 5 = 5 + __

(−2)7 = __(−2)

Use the Associative Properties

When we associate one object with another, we think of those objects as being grouped together.

Addition

Multiplication

( ) ( )a b c a b c

( ) ( )ab c a bc

The associative properties say that when we add or multiply three numbers, we can group the first two together or the last two together and get the same answer.

Use an associative property to complete each statement.

Solution:5 (2 8) ________

10 ( 8) ( 3) ________

(5 2) 8

10 ( 8) ( 3)

Solution:Commutative

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

(2 4)6 (4 2)6

Example of associative and/or commutative property?

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(2 + 4) + 5 = 2 + (4 + 5)

6(3⋅10) = 6(10 ⋅3)

(8 +1) + 7 = 8 + (7 +1)

Use the Identity Properties

If a child wears a costume on Halloween, the child’s appearance is changed, but his or her identity is unchanged.

and Addition

and Multiplication

0a a 0 a a

1a a 1 a a

The identity of a real number is left unchanged when identity properties are applied. The identity properties say:

Use the Inverse Properties

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…OppositesThe inverse properties of addition and

multiplication lead to the additive and multiplicative identities, respectively.

and Addition

and Multiplication

( ) 0a a 0a a 11a

a

11 0)a a

a

Solution:

Complete each statement so that it is an example of an inverse property.

___ 6 0

1___ 1

9

6

9

Use the Distributive Property

The distributive property can be used “in reverse.” For example, we can write .

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

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

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

The distributive property is also valid for multiplication over subtraction.

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

( )a b c d ab ac ad

( )ac bc a b c

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:

Solution:

Write the expression without parentheses.

( 5 8)y 5 8y

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8(3r +11t + 5z)

−2(x + 3)€

=24r + 88t + 40z

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=−2x −6

1.8

Simplifying Expressions

Simplifying Expressions

Simplify each expression.

Solution: 5 4 3x y

7 6 9k

5 4 5 3x y 5 4 5 3x y

20 15x y

) 91(7 6k 1 7 1 6 9k

7 6 9k 7 6 9k

7 9 6k 2 6k

Identify Terms and Numerical Coefficients

A term is a number, a variable, or a product or quotient of numbers and variables raised to powers, such as

, , , , , and . TermsIn the term 9x, the numerical coefficient, or

simply coefficient, of the variable x is 9. In the term −8m2n the numerical coefficient of m2n is −8.

Terms are separated by a + or – If two factors are multiplied together, that is one

term.

9x 215y 3 28m n 2

pk

3 28 12x x 3 28 12x x

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4(3m −2n)

6 + 3(4k + 5)€

=4(3m) − 4(2n)

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=(4 ⋅3)m − (4 ⋅2)n

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=12m −8n

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=6 + 3(4k) + 3(5)

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=6 + (3⋅4)k + 3(5)

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=6 +12k +15

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=6 +15 +12k

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=21+12k

Identify Like Terms

Terms with exactly the same variables that have the same exponents are like terms. For example, 9m and 4m have the same variable and are like terms.

The terms −4y and 4y2 have different exponents and are unlike terms.

5x 12x

24xy 5xy

23x y 25x y

3 37w z 32xz

and andLike terms

andand Unlike terms

Combine Like Terms

Recall the distributive property:

This form of the distributive property may be used to find the sum or difference of like terms.

Using the distributive property in this way is called combining like terms.

( )x y z xy xz

( )xy xz x y z

3 5 (3 5) 8x x x x

This statement can also be written “backward” as

.

Examples

Combine like terms in each expression.

Solution:

5 9 4z z z

4r r

28 8p p

(5 9 4)z 10z

(4 1)r 3r

Cannot be combined

Simplify each expression.Solution:

(3 5 ) 7k k

7 2 (1 )z z

(3 51 ) 7k k 1(3) ( 1)(5 ) 7k k

3 ( 5 ) 7k k 3 2k

7 ( 2) ( 1)(1 )z z

7 ( 2) ( 1)(1) ( 1)( )z z 7 ( 2) ( 1) ( )z z

6 3z

Simplify Expressions from Word Phrases

Translate to a mathematical expression and simplify.

Three times a number, subtracted from the sum of the number and 8.

Solution:( 8) 3x x

8 ( 3 )x x 2 8x

Homework So Far….

1.1 1-91 EOO

1.4 1-73 EOO

1.5 7-113 EOO

1.6 13-113 EOO

1.7 1-79 ODD

1.8 1-83 ODD