Associative & Commutative Property of Addition & Multiplication (With Examples)

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In math, the associative and commutative properties are laws applied to addition and multiplication that always exist. The associative property states that you can re-group numbers and you will get the same answer and the commutative property states that you can move numbers around and still arrive at the same answer.

What is the Associative Property?

The associative property comes from the words "associate" or "group." It refers to grouping of numbers or variables in algebra. You can re-group numbers or variables and you will always arrive at the same answer.

This equation shows the associative property of addition:

(a + b) + c = a + (b + c) \\ (2 + 4) +3 = 2 + (4 + 3)

This equation shows the associative property of multiplication:

(a × b) × c = a × (b × c) \\ (2 × 4) × 3 = 2 × (4 × 3)

In some cases, you can simplify a calculation by multiplying or adding in a different order, but arriving at the same answer:

What is 19 + 36 + 4?

19 + 36 + 4 = 19 + (36 + 4) = 19 + 40 = 59

What is the Commutative Property?

The commutative property in math comes from the words "commute" or "move around." This rule states that you can move numbers or variables in algebra around and still get the same answer.

This equation defines the commutative property of addition:

a + b = b + a \\ 4 + 2 = 2 + 4

This equation defines commutative property of multiplication:

a × b = b × a \\ 3 × 2 = 2 × 3

Sometimes rearranging the order makes it easier to add or multiply:

What is 2 × 16 × 5?

2 × 16 × 5 = (2 × 5) × 16 = 10 × 16 = 160

 Additional Practice Problems for Students

6 + (4 + 2) = 12 \text{ so } (6 + 4) + 2 =

Find the missing number in this equation:

3 + ( \_ + 5) = (3 + 7) + 5

What is this equation equal to:

6 × (2 × 9) = ?

Find the missing number:

2 + (\_ + 4) = (2 + 8) + 4

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