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 