Carlos makes salad dressings with olive oil and balsamic vinegar. Sometimes he adds the olive oil first and other times he adds the vinegar first. The salad dressing is always the same, so the order doesn’t matter.
The order in which any two numbers are either added or multiplied doesn’t change the sum or product. Addition and multiplication are said to be commutative.
Example 1: 3x ! 4y ! 5x
Commutative Property For any two numbers of Addition a and b, a ! b " b ! a.
Commutative Property For any two numbers of Multiplication a and b, a $ b " b $ a.
" 3x ! 5x ! 4y Commutative (!)
" 8x ! 4y
Example 2: 5 # 11 # 2
" 5 # 2 # 11 Commutative (")
" 10 # 11
" 110
You can also regroup numbers when you are adding or multiplying without changing the sum or product. Addition and multiplication are said to be associative.
Example 3: (10 ! 5) ! 8
Associative Property For any numbers a, b, and c, of Addition (a ! b) ! c " a ! (b ! c).
Associative Property For any numbers a, b, and c, of Multiplication (a $ b) $ c " a $ (b $ c).
" 10 ! (5 ! 8) Associative (!)
" 23
Example 4: (100 $ 4) $ 5
" 100 $ (4 $ 5) Associative (")
" 2000
Simplify each expression.
1. (d ! 7) ! 3 d ! 10 2. 2x ! 7 ! 5x 7x ! 7 3. 2 # 7k # 5 70k
4. (4a ! 2b) ! (a ! b) 5. 7 $ y $ 3 21y 6. (8 # m) # 4 32m 5a ! 3b
Name the property shown by each statement.
7. 29 ! b " b ! 29 Commutative 8. 2(4 $ 6) " (2 $ 4)6 Associative Property of Addition Property of Multiplication 9. (3 ! 21) ! 7 " (3 ! 7) ! 21 10. 42 $ 8 " 8 $ 42 Associative Property Commutative Property of Addition of Multiplication