# Commutative and Associative Properties

## Commutative and Associative Properties

#### Use the Commutative and Associative Properties

Think about adding two numbers, such as 5 and 3 .

Notice, the order in which we add does not matter. The same is true when multiplying 5 and 3.

Again, the results are the same! . The order in which we multiply does not matter. These examples illustrate the commutative properties of addition and multiplication.

##### COMMUTATIVE PROPERTIES

Commutative Property of Addition: if and are real numbers, thenCommutative Property of Multiplication: if and are real numbers, then

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

What about subtraction? Does order matter when we subtract numbers? Does give the same result as

Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.

Let's see what happens when we divide two numbers. Is division commutative?

Since changing the order of the division did not give the same result, division is not commutative.

Addition and multiplication are commutative. Subtraction and division are not commutative.

Suppose you were asked to simplify this expression.

How would you do it and what would your answer be?

Some people would think is 15 and then is . Others might start with makes 10 and then makes .

Both ways give the same result, as shown in Figure 7.3. (Remember that parentheses are grouping symbols that indicate which operations should be done first).

**Figure 7.3**

When adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.

The same principle holds true for multiplication as well. Suppose we want to find the value of the following expression:

Changing the grouping of the numbers gives the same result, as shown in Figure 7.4.

Figure 7.4.

When multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.

If we multiply three numbers, changing the grouping does not affect the product.

You probably know this, but the terminology may be new to you. These examples illustrate the *Associative Properties*.

**ASSOCIATIVE PROPERTIES**

**Associative Property of Addition:** if , and are real numbers, then

**Associative Property of Multiplication:** if , and are real numbers, then

Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.

#### Evaluate Expressions using the Commutative and Associative Properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.

Source: Rice University, https://openstax.org/books/prealgebra/pages/7-2-commutative-and-associative-properties

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