Commutative and Associative Properties


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The commutative property states that the order in which you add or multiply numbers does not change their sum or product.

For Example:

3 + 5 = 5 + 3
a + b = b + a

6 · 5 = 5 · 6
ab = ba



The associative property says the when adding or multiplying 3 or more numbers or expressions together, the way in which you group the numbers or expressions does not change the final sum or product.

For Example:

(4 + 5) + 7 = 4 + (5 + 7)
(a + b) + c = a + (b + c)

(2 · 7) · 4 = 2 · (7 · 4)
(ab)c = a(bc)


Using Commutative and Associative Properties to Simplify Expressions

The associative and commutative properties can be used along with other properties of numbers to help simplify algebraic expressions. Let's take a look at the following expression:

3x + 2y + 5x

In this example we can use the commutative property to rearrange the terms being added together.

3x + 5x + 2y

We can add like terms together because of the distributive property.

3x + 5x = (3 + 5)x = 8x

Because of the distributive property we can now just add like terms by adding coefficients.

3x + 5x + 2y = 8x + 2y



Let's now look at another example:

Simplify 2(a + b) + 6(a + 5b) :


2(a + b) + 3(a + 2b) = 2a + 2b + 3a + 6b Distributive Property
= 2a + 6a + 3b + 6b Commutative Property
= (2a + 6a) + (3b + 6b) Associative Property
= (2 + 6)a + (3 + 6)b Distributive Property
= 8a + 9b Distributive Property


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