In algebra, we will often need to simplify an expression to make it easier to use. We are going to review some important properties of algebra that makes simplification possible.
Commutative Properties of Addition and Multiplication
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:
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.
Addition and multiplication is commutative. When adding terms, you can add them in any order you wish. When multiplying terms, you may multiply the terms in any order you wish.
Associative Properties of Addition and Multiplication
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:
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.
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:
Notice that in the previous example, we multiplied each term inside the parentheses by a negative number. With the subtraction inside, this means we multiplied -7 by -6, to result in positive 42. The most common error in distributing is a sign error. Be careful with your signs!
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html