1. Real Numbers
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2. Properties of Real Numbers
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Properties of Real Numbers
Use the commutative properties.
Use the associative properties.
Use the identity properties.
Use the inverse properties.
Use the distributive properties. 2 3 4 5

3. 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.
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4. Use the commutative properties.
Objective 1
Use the commutative properties.
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5. Commutative Properties
Use the commutative properties.
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
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6. Using the Commutative Properties
EXAMPLE 1
Using the Commutative Properties
Use a commutative property to complete each statement.
Solution:
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7. Use the associative properties.
Objective 2
Use the associative properties.
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8. Associative Properties
Use the associative properties.
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
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9. Using the Associative Properties
EXAMPLE 2
Using the Associative Properties
Use an associative property to complete each statement.
Solution:
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10. Distinguishing Between Properties
EXAMPLE 3
Distinguishing Between Properties
Is an example of the associative property or the commutative property?
Solution:
Commutative
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11. Using the Commutative and Associative Properties
EXAMPLE 4
Using the Commutative and Associative Properties
Find the sum.
Solution:
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12. Use the identity properties.
Objective 3
Use the identity properties.
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13. Identity Properties Addition Multiplication
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. Likewise, the identity of a real number is left unchanged when identity properties are applied.
Identity Properties
Addition
Multiplication
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.
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14. Using the Identity Properties
EXAMPLE 5
Using the Identity Properties
Use an identity property to complete each statement.
Solution:
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15. Using the Identity Property to Simplify Expressions
EXAMPLE 6
Using the Identity Property to Simplify Expressions
Simplify.
Solution:
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16. Use the inverse properties.
Objective 4
Use the inverse properties.
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17. Inverse Properties Addition Multiplication
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.
Inverse Properties
Addition
Multiplication
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18. Using the Inverse Properties
EXAMPLE 7
Using the Inverse Properties
Use an inverse property to complete each statement.
Solution:
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19. Using Properties to Simplify an Expression
EXAMPLE 8
Using Properties to Simplify an Expression
Simplify.
Solution:
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20. Use the distributive property.
Objective 5
Use the distributive property.
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21. Using the Distributive Property
EXAMPLE 9
Using the Distributive Property
Use the distributive property to rewrite each expression.
Solution:
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22. Use the distributive properties.
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,
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23. Distributive Property
Use the distributive properties. (cont’d)
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.
Distributive Property
The distributive property is also valid for multiplication over subtraction.
and
The distributive property can be extended to more than two numbers.
The distributive property can be used “in reverse.”
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24. Solution: EXAMPLE 10 Write the expression without parentheses.
Using the Distributive Property to Remove (Clear) Parentheses
Write the expression without parentheses.
Solution:
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