PROPERTIES OF ADDITION AND MULTIPLICATION

Commutative Property

Words :

You can add numbers in any order and multiply numbers in any order.

Numbers : 

3 + 8  =  8 + 3  

7 ⋅ 4  =  4 ⋅ 7

Algebra : 

x + y  =  y + x

xy  =  yx

Associative Property

Words :

When you are only adding or only multiplying, you can group any of the numbers together. 

Numbers : 

(7 + 9) + 3  =  7 + (9 + 3)  

(6 ⋅ 8) ⋅ 2  =  6 ⋅ (8 ⋅ 2)

Algebra : 

(x + y) + z  =  x + (y + z)

(xy)z  =  x(yz)

Simplifying Expressions

Simplify each expression. 

Problem 1 : 

35 + 43 + 65

Solution : 

=  35 + 43 + 65

Use the Commutative Property of Addition.

=  (35 + 65) + 43

Use the Associative Property of Addition to make groups of compatible numbers.

=  100 + 43

=  143

Problem 2 : 

4 ⋅ 7 ⋅ 25

Solution : 

=  4 ⋅ 7 ⋅ 25

Use the Commutative Property of Multiplication.

=  7 ⋅ 4 ⋅ 25

Use the Associative Property of Multiplication to make groups of compatible numbers.

=  7 ⋅ (4 ⋅ 25)

=  7 ⋅ 100

=  700

Problem 3 : 

15⅓ + 4 + 1⅔

Solution : 

=  15⅓ + 4 + 1⅔

Use the Commutative Property of Addition.

=  15⅓ + 1⅔ + 4

Use the Associative Property of Addition to make groups of compatible numbers.

=  (15⅓ + 1⅔) + 4

=  (46/3 + 5/3) + 4

=  51/3 + 4

=  17 + 4

=  21

Problem 4 : 

8 ⋅ 5 ⋅ 3/2

Solution : 

=   8 ⋅ 5 ⋅ 3/2

Use the Commutative Property of Multiplication.

=  5 ⋅ 8 ⋅ 3/2

Use the Associative Property of Multiplication to make groups of compatible numbers.

=  5 ⋅ (8 ⋅ 3/2)

=  5 ⋅ 24/2

=  5 ⋅ 12

=  60

Problem 5 : 

410 + 58 + 90 + 2

Solution : 

=   410 + 58 + 90 + 2

Use the Commutative Property of Addition.

=  410 + 90 + 58 + 2

Use the Associative Property of Addition to make groups of compatible numbers.

=  (410 + 90) + (58 + 2)

=  500 + 60

=  560

Problem 6 : 

2.5 ⋅ 3 ⋅ 4 ⋅ 6

Solution : 

=   2.5 ⋅ 3 ⋅ 4 ⋅ 6

Use the Commutative Property of Multiplication.

=   2.5 ⋅ 4 ⋅ 3 ⋅ 6

Use the Associative Property of Multiplication to make groups of compatible numbers.

=  (2.5 ⋅ 4) ⋅ (3 ⋅ 6)

 =  10 + 18

=  28

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