In short, the commutative properties of addition and multiplication allow us to add algebraic terms in any order we wish, as well as multiply algebraic terms in any order we wish. These properties are illustrated in the following examples:

To simplify, add 4x to 3x  

Sum of 3x and 4x  

To simplify, add 3x to 4x  

Sum of 3x and 4x  
 

To simplify, multiply 5 by 2c  

Product of 5 and 2c  

To simplify, multiply 2c by 5  

Product of 2c and 5 
In each example above, notice that the order in which applied the operation (either addition or multiplication) did not affect the solution. It is important to note that subtraction and division are not commutative.
The associative property deals with how terms of an expression are grouped together. For algebraic expressions in which several terms are being added together, you can group terms together in any way in order to make simplification easier. Here is an example of the associative property of addition:

We can group terms in any way  

Add 2a to 5a  

Our Solution 
In some cases, such as the example above, the associative property is helpful when grouping like terms together. We used the associative property first add 2a and 5a to get 7a, then we added 3 at the end.
The associative property holds true for multiplication as well, and works in a similar way to addition.

We can group terms in any way  

Multiply x and 3x  

Multiply by 2  

Our Solution 
Often as we work with problems, there will be a set of parentheses that make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses, we can use the distributive property. Using this property, we multiply the number in front of the parentheses by each term inside. Here are some examples:

Multiply each term by 4  

Our Solution  
 

Multiply each term by  

Our Solution 
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html