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Properties in Algebraic Expressions

Properties in Algebraic Expressions

Author: Sophia Tutorial

Simplify an algebraic expression using the Distributive Property.

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what's covered
  1. Commutative Properties of Addition and Multiplication
  2. Associative Properties of Addition and Multiplication
  3. Distributive Property

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:

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

5 times 2 c
To simplify, multiply 5 by 2c
10 c
Product of 5 and 2c
2 c times 5
To simplify, multiply 2c by 5
10 c
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 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.

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. Here is an example of the associative property of addition:

open parentheses 3 plus 2 a close parentheses plus 5 a
We can group terms in any way
3 plus open parentheses 2 a plus 5 a close parentheses
Add 2a to 5a
3 plus 7 a
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.

x times open parentheses 3 x times 2 close parentheses
We can group terms in any way
open parentheses x times 3 x close parentheses times 2
Multiply x and 3x
open parentheses 3 x squared close parentheses 2
Multiply by 2
6 x squared
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. Here are some examples:

4 open parentheses 2 x minus 7 close parentheses
Multiply each term by 4
8 x minus 28
Our Solution

short dash 7 open parentheses 5 x minus 6 close parentheses
Multiply each term by short dash 7
short dash 35 x plus 42
Our Solution

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!

The commutative, associative, distributive properties, as well as factoring, can be extended to expressions that are involving variables. These properties will be useful when simplifying expressions and solving equations.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: