Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

Properties of Addition / Properties of Multiplication
Commutative Property
For any numbers a and b:



(Order in which we add does not matter) / Commutative Property
For any numbers a and b:



(Order in which we multiply does not matter)
Associative Property
For any numbers a, b, and c:



(Grouping numbers during an addition process is okay)
Note: The positions of the numbers DO NOT change. / Associative Property
For any numbers a, b, and c:



(Grouping numbers during a multiplication process is okay)
Note: The positions of the numbers DO NOT change.
Identity Element
There exists a real number 0 such that for every real a
,

(Technical, definition and use of “0”.) / Identity Element
There exists a real number 1 such that for every real a
,

(Technical, definition and use of “1”.)
Additive Inverse (Opposite)
For every real number a there exists a real number, denoted (–a), such that


(Definition of the term “opposite”.) / Multiplicative Inverse (Reciprocal)
For every real number a except 0 there exists a real number, denoted, such that


(Definition of the term “reciprocal”.)
Distributive Property (of Multiplication over Addition/Subtraction)
For any numbers a, b, and c:
,
,
,
(Do not confuse this with the “Associative Property”. “Distributive Property” has two different operations: “”& “+”)
Note: We can distribute from the left or the right because of the “commutativity” of multiplication.

The commutative and associative properties DO NOT hold for subtraction and division. That is, subtraction and division are NOT commutative and NOT associative. Hence:

is not equal to
is not equal to
is not equal to
is not equal to / is not equal to
is not equal to
is not equal to
is not equal to

These properties mean that order and grouping (association) don’t matter for addition and multiplication, but they certainly do matter for subtraction and division. In this way, addition and multiplication are “cleaner” than subtraction and division.

 Exercises:

Use the commutative law of addition to write an equivalent expression.

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

1)a + 2

2)3a + 7b

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

Use the commutative law of multiplication to write an equivalent expression.

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

3)xy

4)9(x + 5)

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

Use the associative law of addition to write an equivalent expression.

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

5)(5 + m) + r

6)(m + np) + b

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

Use the commutative law of multiplication to write an equivalent expression.

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

7)(9a)b

8)5[x(2 + y)]

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

Multiply.

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

9)3(x + 5)

11)6(v + 4)

13)9(x + 3)

10)9(6m +7)

12)4(5x + 8 + 3p)

14)(x + 2)7

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

Use the distributive law to factor each of the following. Check by multiplying.

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

15)5y + 5z

17)20a + 5

16)13 + 13x

18)14a + 56b + 7

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Math 010 - CooleyElementary Algebra OCC

Section 1.2 – The Commutative, Associative, and Distributive Laws

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