Properties in Algebraic Expressions

1. 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:

EXAMPLE

	To simplify, add 4x to 3x
	Sum of 3x and 4x
	To simplify, add 3x to 4x
	Sum of 3x and 4x

EXAMPLE

	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.

big idea

Addition and multiplication are 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.

2. 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.

EXAMPLE

	We can group terms in any way. Regroup and instead
	Add to
	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 to add and to get then we added 3 at the end.

The associative property holds true for multiplication as well and works in a similar way to addition.

EXAMPLE

	We can group terms in any way. Regroup x and 3x
	Multiply x and 3x
	Multiply by 2
	Our Solution

3. Distributive Property

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.

EXAMPLE

	Multiply each term by 4
	Our Solution

EXAMPLE

	Multiply each term by
	Our Solution

hint

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 a positive 42. The most common error in distributing is a sign error. Be careful with your signs!