Commutative and Associative Properties

Site: Saylor Academy
Course: RWM101: Foundations of Real World Math
Book: Commutative and Associative Properties
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Date: Saturday, April 13, 2024, 1:56 AM


Read this section on the communitive and associative properties. Complete the practice questions and check your answers.

Simplify Expressions Using the Commutative and Associative Properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in Example 7.8 part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is 0.0. Likewise, part ⓑ in Example 7.9 was easier, with the reciprocals grouped together, because their product is 1.1. In the next few examples, we'll use our number sense to look for ways to apply these properties to make our work easier.

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals - their product is 

In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.

When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead.

When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.

In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression 3 x+7+4 x+5 by rewriting it as 3 x+4 x+7+5 and then simplified it to 7 x+12. We were using the Commutative Property of Addition.

Source: Rice University,
Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 License.

Examples and Exercises


Use the associative properties to rewrite the following:

(a) (3+0.6)+0.4= ________

(b) \left(-4 \cdot \frac{2}{5}\right) \cdot 15= ________

Try It 7.11

Use the associative properties to rewrite the following:

(a) (1+0.7)+0.3= \text{________}                                                  (b) (-9 \cdot 8) \cdot \frac{3}{4}= \text{________}

Try It 7.12

Use the associative properties to rewrite the following:

(a) (4+0.6)+0.4= \text{________}                                                  (b) (-2 \cdot 12) \cdot \frac{5}{6}= \text{________}


Solution to Example 7.6

Change the grouping.


  \left(-4 \cdot \frac{2}{5}\right) \cdot 15= _______
Change the grouping. \left(-4 \cdot \frac{2}{5}\right) \cdot 15=-4 \cdot\left(\frac{2}{5} \cdot 15\right)

Notice that \frac{2}{5}⋅15 is 6. The multiplication will be easier if we group as shown on the right.

Try It 7.11

(a) (1+0.7)+0.3=1+(0.7+0.3)

(b) (-9 \cdot 8) \cdot \frac{3}{4}=-9\left(8 \cdot \frac{3}{4}\right)

Try It 7.12

(a) (4+0.6)+0.4=4+(0.6+0.4)

(b) (-2 \cdot 12) \cdot \frac{5}{6}=-2\left(12 \cdot \frac{5}{6}\right)