Download Section 1.2 - The Commutative, Associative, and

Math 010 - Cooley
Elementary 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:
abba
3 443
77
(Order in which we add does not matter)
Associative Property
For any numbers a, b, and c:
a  b  c   a  b   c
2  3  4  2  3  4
99
Commutative Property
For any numbers a and b:
a b  ba
3 4  43
12  12
(Order in which we multiply does not matter)
Associative Property
For any numbers a, b, and c:
a  b  c   a  b   c
2  3  4  2  3  4
24  24
(Grouping numbers during an addition process is okay)
(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
a0a , 0aa
50055
(Technical, definition and use of “0”.)
Note: The positions of the numbers DO NOT change.
Identity Element
There exists a real number 1 such that for every real a
a 1  a , 1  a  a
5 1  1 5  5
(Technical, definition and use of “1”.)
Multiplicative Inverse (Reciprocal)
For every real number a except 0 there exists a real
number, denoted 1a , such that
Additive Inverse (Opposite)
For every real number a there exists a real number,
denoted (–a), such that
a   a   0
3   3  0
(Definition of the term “opposite”.)
a  1a  1
3  13  1
(Definition of the term “reciprocal”.)
Distributive Property (of Multiplication over Addition/Subtraction)
For any numbers a, b, and c:
ab  c   ab  ac , ab  c   ab  ac
a  b c  ac  bc , a  b c  ac  bc
2x  3  2 x  6 , ( y  4)(3)  3 y  12
(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:
ab
a b

a  b  c
a  b  c 
is not equal to
is not equal to
is not equal to
is not equal to
ba
ba
a  b   c
a  b   c
2  3  1 is not equal to 3  2  1
1  2  12 is not equal to 2  1  2
2  3  4  3 is not equal to 2  3  4  5
2  3  4  83 is not equal to 2  3  4  16
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.
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Math 010 - Cooley
Elementary Algebra
OCC
Section 1.2 – The Commutative, Associative, and Distributive Laws
 Exercises:
Use the commutative law of addition to write an equivalent expression.
1)
a+2
2)
3a + 7b
Use the commutative law of multiplication to write an equivalent expression.
3)
xy
4)
9(x + 5)
Use the associative law of addition to write an equivalent expression.
5)
(5 + m) + r
6)
(m + np) + b
Use the commutative law of multiplication to write an equivalent expression.
7)
(9a)b
8)
5[x(2 + y)]
Multiply.
9)
3(x + 5)
10)
9(6m +7)
11)
6(v + 4)
12)
4(5x + 8 + 3p)
13)
9(x + 3)
14)
(x + 2)7
Use the distributive law to factor each of the following. Check by multiplying.
15)
5y + 5z
16)
13 + 13x
17)
20a + 5
18)
14a + 56b + 7
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