Learning Objectives
 Use the commutative and associative properties
 Evaluate expressions using the commutative and associative properties
 Simplify expressions using the commutative and associative properties
Before you get started, take this readiness quiz.
Simplify: $7y+2+y+13.$
If you missed this problem, review Example 2.22.
Multiply: $\frac{2}{3}\xb718.$
If you missed this problem, review Example 4.28.
Find the opposite of $15.$
If you missed this problem, review Example 3.3.
In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations in the next chapter.
Use the Commutative and Associative Properties
Think about adding two numbers, such as $5$ and $3.$
The results are the same. $5+3=3+5$
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! $5\xb73=3\xb75.$ 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 $a$ and $b$ are real numbers, then
Commutative Property of Multiplication: if $a$ and $b$ 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.
Example 7.5
Use the commutative properties to rewrite the following expressions:

ⓐ$\phantom{\rule{0.2em}{0ex}}\mathrm{1}+3=\_\_\_\_\_$

ⓑ$\phantom{\rule{0.2em}{0ex}}4\xb79=\_\_\_\_\_$
ⓐ  
$\mathrm{1}+3=\_\_\_\_\_$  
Use the commutative property of addition to change the order.  $\mathrm{1}+3=3+(\mathrm{1})$ 
ⓑ  
$4\xb79=\_\_\_\_\_$  
Use the commutative property of multiplication to change the order.  $4\xb79=9\xb74$ 
Use the commutative properties to rewrite the following:
 ⓐ$\phantom{\rule{0.2em}{0ex}}\mathrm{4}+7=\_\_\_\_\_\phantom{\rule{0.2em}{0ex}}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}6\xb712=\_\_\_\_\_$
Use the commutative properties to rewrite the following:
 ⓐ$\phantom{\rule{0.2em}{0ex}}14+(\mathrm{2})=\_\_\_\_\_\phantom{\rule{0.2em}{0ex}}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}3(\mathrm{5})=\_\_\_\_\_$
What about subtraction? Does order matter when we subtract numbers? Does $73$ give the same result as $37?$
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 $7+8\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}15$ and then $15+2\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}17.$ Others might start with $8+2\phantom{\rule{0.2em}{0ex}}\text{makes}\phantom{\rule{0.2em}{0ex}}10$ and then $7+10\phantom{\rule{0.2em}{0ex}}\text{makes}\phantom{\rule{0.2em}{0ex}}17.$
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.)
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.
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 $a,b,$ and $c$ are real numbers, then
Associative Property of Multiplication: if $a,b,$ and $c$ are real numbers, then
Example 7.6
Use the associative properties to rewrite the following:

ⓐ$\phantom{\rule{0.2em}{0ex}}(3+0.6)+0.4=\_\_\_\_\_\_\_\_\_\_$

ⓑ$\phantom{\rule{0.2em}{0ex}}\left(\mathrm{4}\xb7\frac{2}{5}\right)\xb715=\_\_\_\_\_\_\_\_\_\_$
ⓐ  
$(3+0.6)+0.4=\_\_\_\_\_\_\_\_\_\_$  
Change the grouping.  $(3+0.6)+0.4=3+(0.6+0.4)$ 
Notice that $0.6+0.4$ is $1,$ so the addition will be easier if we group as shown on the right.
ⓑ  
$\left(\mathrm{4}\xb7\frac{2}{5}\right)\xb715=\_\_\_\_\_\_\_\_\_\_$  
Change the grouping.  $\left(\mathrm{4}\xb7\frac{2}{5}\right)\xb715=\mathrm{4}\xb7(\frac{2}{5}\xb715)$ 
Notice that $\frac{2}{5}\xb715$ is $6.$ The multiplication will be easier if we group as shown on the right.
Use the associative properties to rewrite the following:
ⓐ $(1+0.7)+0.3=\_\_\_\_\_\_\_\_\_\_\phantom{\rule{2em}{0ex}}$ ⓑ$\phantom{\rule{0.2em}{0ex}}(\mathrm{9}\xb78)\xb7\frac{3}{4}=\_\_\_\_\_\_\_\_\_\_$
Use the associative properties to rewrite the following:
ⓐ $(4+0.6)+0.4=\_\_\_\_\_\_\_\_\_\_\phantom{\rule{2em}{0ex}}$ ⓑ$\phantom{\rule{0.2em}{0ex}}(\mathrm{2}\xb712)\xb7\frac{5}{6}=\_\_\_\_\_\_\_\_\_\_$
Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.
Example 7.7
Use the Associative Property of Multiplication to simplify: $6(3x).$
$6(3x)$  
Change the grouping.  $(6\xb73)x$ 
Multiply in the parentheses.  $18x$ 
Notice that we can multiply $6\xb73,$ but we could not multiply $3\xb7x$ without having a value for $x.$
Use the Associative Property of Multiplication to simplify the given expression: $8(4x).$
Use the Associative Property of Multiplication to simplify the given expression: $\mathrm{9}(7y).$
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 regroup terms to make our work easier, as the next several examples illustrate.
Example 7.8
Evaluate each expression when $x=\frac{7}{8}.$
 ⓐ$\phantom{\rule{0.2em}{0ex}}x+0.37+(x)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}x+(x)+0.37$
ⓐ  
Substitute $\frac{7}{8}$ for $x$.  
Convert fractions to decimals.  
Add left to right.  
Subtract. 
ⓑ  
Substitute $\frac{7}{8}$ for x.  
Add opposites first. 
What was the difference between part ⓐ and part ⓑ? Only the order changed. By the Commutative Property of Addition, $x+0.37+(x)=x+(x)+0.37.$ But wasn’t part ⓑ much easier?
Evaluate each expression when $y=\frac{3}{8}:\phantom{\rule{0.2em}{0ex}}$ⓐ$\phantom{\rule{0.2em}{0ex}}y+0.84+(y)$ ⓑ$\phantom{\rule{0.2em}{0ex}}y+(y)+0.84.$
Evaluate each expression when $f=\frac{17}{20}:\phantom{\rule{0.2em}{0ex}}$ⓐ$\phantom{\rule{0.2em}{0ex}}f+0.975+(f)$ ⓑ $\phantom{\rule{0.2em}{0ex}}f+(f)+0.975.$
Let’s do one more, this time with multiplication.
Example 7.9
Evaluate each expression when $n=17.$

ⓐ$\phantom{\rule{0.2em}{0ex}}\frac{4}{3}\left(\frac{3}{4}n\right)$

ⓑ$\phantom{\rule{0.2em}{0ex}}\left(\frac{4}{3}\xb7\frac{3}{4}\right)n$
ⓐ  
Substitute 17 for n.  
Multiply in the parentheses first.  
Multiply again. 
ⓑ  
Substitute 17 for n.  
Multiply. The product of reciprocals is 1.  
Multiply again. 
What was the difference between part ⓐ and part ⓑ here? Only the grouping changed. By the Associative Property of Multiplication, $\frac{4}{3}\left(\frac{3}{4}n\right)=\left(\frac{4}{3}\xb7\frac{3}{4}\right)n.$ By carefully choosing how to group the factors, we can make the work easier.
Evaluate each expression when $p=24\text{:}\phantom{\rule{0.2em}{0ex}}$ⓐ$\phantom{\rule{0.2em}{0ex}}\frac{5}{9}\left(\frac{9}{5}p\right)\phantom{\rule{2em}{0ex}}$ ⓑ$\phantom{\rule{0.2em}{0ex}}\left(\frac{5}{9}\xb7\frac{9}{5}\right)p.$
Evaluate each expression when $q=15\text{:}\phantom{\rule{0.2em}{0ex}}$ⓐ$\phantom{\rule{0.2em}{0ex}}\frac{7}{11}\left(\frac{11}{7}q\right)\phantom{\rule{2em}{0ex}}$ ⓑ$\phantom{\rule{0.2em}{0ex}}(\frac{7}{11}\xb7\frac{11}{7})q$
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 $0.$ Likewise, part ⓑ in Example 7.9 was easier, with the reciprocals grouped together, because their product is $1.$ In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.
Example 7.10
Simplify: $\mathrm{84}n+(\mathrm{73}n)+84n.$
Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.
$\mathrm{84}n+(\mathrm{73}n)+84n$  
Reorder the terms.  $\mathrm{84}n+84n+(\mathrm{73}n)$ 
Add left to right.  $0+(\mathrm{73}n)$ 
Add.  $\mathrm{73}n$ 
Simplify: $\mathrm{27}a+(\mathrm{48}a)+27a.$
Simplify: $39x+(\mathrm{92}x)+(\mathrm{39}x).$
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is $1.$
Example 7.11
Simplify: $\frac{7}{15}\xb7\frac{8}{23}\xb7\frac{15}{7}.$
Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.
$\frac{7}{15}\xb7\frac{8}{23}\xb7\frac{15}{7}$  
Reorder the terms.  $\frac{7}{15}\xb7\frac{15}{7}\xb7\frac{8}{23}$ 
Multiply left to right.  $1\xb7\frac{8}{23}$ 
Multiply.  $\frac{8}{23}$ 
Simplify: $\frac{9}{16}\xb7\frac{5}{49}\xb7\frac{16}{9}.$
Simplify: $\frac{6}{17}\xb7\frac{11}{25}\xb7\frac{17}{6}.$
In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.
Example 7.12
Simplify: $\left(\frac{5}{13}+\frac{3}{4}\right)+\frac{1}{4}.$
Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.
$\left(\frac{5}{13}+\frac{3}{4}\right)+\frac{1}{4}$  
Group the terms with a common denominator.  $\frac{5}{13}+\left(\frac{3}{4}+\frac{1}{4}\right)$ 
Add in the parentheses first.  $\frac{5}{13}+\left(\frac{4}{4}\right)$ 
Simplify the fraction.  $\frac{5}{13}+1$ 
Add.  $1\frac{5}{13}$ 
Convert to an improper fraction.  $\frac{18}{13}$ 
Simplify: $\left(\frac{7}{15}+\frac{5}{8}\right)+\frac{3}{8}.$
Simplify: $\left(\frac{2}{9}+\frac{7}{12}\right)+\frac{5}{12}.$
When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.
Example 7.13
Simplify: $(6.47q+9.99q)+1.01q.$
Notice that the sum of the second and third coefficients is a whole number.
$(6.47q+9.99q)+1.01q$  
Change the grouping.  $6.47q+(9.99q+1.01q)$ 
Add in the parentheses first.  $6.47q+(11.00q)$ 
Add.  $17.47q$ 
Many people have good number sense when they deal with money. Think about adding $99$ cents and $1$ cent. Do you see how this applies to adding $9.99+1.01?$
Simplify: $(5.58c+8.75c)+1.25c.$
Simplify: $(8.79d+3.55d)+5.45d.$
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.
Example 7.14
Simplify the expression: $\left[1.67(8)\right](0.25).$
Notice that multiplying $(8)(0.25)$ is easier than multiplying $1.67(8)$ because it gives a whole number. (Think about having $8$ quarters—that makes $\text{\$2.)}$
$\left[1.67(8)\right](0.25)$  
Regroup.  $1.67\left[(8)(0.25)\right]$ 
Multiply in the brackets first.  $1.67\left[2\right]$ 
Multiply.  $3.34$ 
Simplify: $\left[1.17(4)\right](2.25).$
Simplify: $\left[3.52(8)\right](2.5).$
When simplifying expressions that contain variables, we can use the commutative and associative properties to reorder or regroup terms, as shown in the next pair of examples.
Example 7.15
Simplify: $6(9x).$
$6(9x)$  
Use the associative property of multiplication to regroup.  $(6\xb79)x$ 
Multiply in the parentheses.  $54x$ 
Simplify: $8(3y).$
Simplify: $12(5z).$
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 $3x+7+4x+5$ by rewriting it as $3x+4x+7+5$ and then simplified it to $7x+12.$ We were using the Commutative Property of Addition.
Example 7.16
Simplify: $18p+6q+(\mathrm{15}p)+5q.$
Use the Commutative Property of Addition to reorder so that like terms are together.
$18p+6q+(\mathrm{15}p)+5q$  
Reorder terms.  $18p+(\mathrm{15}p)+6q+5q$ 
Combine like terms.  $3p+11q$ 
Simplify: $23r+14s+9r+(\mathrm{15}s).$
Simplify: $37m+21n+4m+(\mathrm{15}n).$
Links To Literacy
Section 7.2 Exercises
Practice Makes Perfect
Use the Commutative and Associative Properties
In the following exercises, use the commutative properties to rewrite the given expression.
$8+9=\_\_\_$
$8(\mathrm{12})=\_\_\_$
$(\mathrm{19})(\mathrm{14})=\_\_\_$
$\mathrm{11}+8=\_\_\_$
$x+4=\_\_\_$
$\mathrm{2}a=\_\_\_$
In the following exercises, use the associative properties to rewrite the given expression.
$(11+9)+14=\_\_\_$
$(12\xb75)\xb77=\_\_\_$
$(\mathrm{7}+9)+8=\_\_\_$
$\left(16\xb7\frac{4}{5}\right)\xb715=\_\_\_$
$3(4x)=\_\_\_$
$(12+x)+28=\_\_\_$
Evaluate Expressions using the Commutative and Associative Properties
In the following exercises, evaluate each expression for the given value.
If $y=\frac{5}{8},$ evaluate:
 ⓐ$\phantom{\rule{0.2em}{0ex}}y+0.49+(y)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}y+(y)+0.49$
If $z=\frac{7}{8},$ evaluate:
 ⓐ$\phantom{\rule{0.2em}{0ex}}z+0.97+(z)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}z+(z)+0.97$
If $c=\frac{11}{4},$ evaluate:
 ⓐ$\phantom{\rule{0.2em}{0ex}}c+3.125+(c)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}c+(c)+3.125$
If $d=\frac{9}{4},$ evaluate:
 ⓐ$\phantom{\rule{0.2em}{0ex}}d+2.375+(d)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}d+(d)+2.375$
If $j=11,$ evaluate:
 ⓐ$\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\left(\frac{6}{5}j\right)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}\left(\frac{5}{6}\xb7\frac{6}{5}\right)j$
If $k=21,$ evaluate:
 ⓐ$\phantom{\rule{0.2em}{0ex}}\frac{4}{13}\left(\frac{13}{4}k\right)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}\left(\frac{4}{13}\xb7\frac{13}{4}\right)k$
If $m=\mathrm{25},$ evaluate:
 ⓐ$\phantom{\rule{0.2em}{0ex}}\frac{3}{7}\left(\frac{7}{3}m\right)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}\left(\frac{3}{7}\xb7\frac{7}{3}\right)m$
If $n=\mathrm{8},$ evaluate:
 ⓐ$\phantom{\rule{0.2em}{0ex}}\frac{5}{21}\left(\frac{21}{5}n\right)$
 ⓑ$\phantom{\rule{0.2em}{0ex}}\left(\frac{5}{21}\xb7\frac{21}{5}\right)n$
Simplify Expressions Using the Commutative and Associative Properties
In the following exercises, simplify.
$\mathrm{45}a+15+45a$
$\frac{1}{2}+\frac{7}{8}+\left(\frac{1}{2}\right)$
$\frac{3}{20}\xb7\frac{49}{11}\xb7\frac{20}{3}$
$\frac{7}{12}\xb7\frac{9}{17}\xb7\frac{24}{7}$
$\mathrm{24}\xb77\xb7\frac{3}{8}$
$\left(\frac{5}{6}+\frac{8}{15}\right)+\frac{7}{15}$
$\frac{5}{13}+\frac{3}{4}+\frac{1}{4}$
$(4.33p+1.09p)+3.91p$
$17(0.25)(4)$
$\left[2.48(12)\right](0.5)$
$7(4a)$
$\mathrm{15}(5m)$
$12\left(\frac{5}{6}p\right)$
$14x+19y+25x+3y$
$43m+(\mathrm{12}n)+\text{}(\mathrm{16}m)+(\mathrm{9}n)$
$\frac{3}{8}g+\frac{1}{12}h+\frac{7}{8}g+\frac{5}{12}h$
$6.8p+9.14q+(\mathrm{4.37}p)+(\mathrm{0.88}q)$
Everyday Math
Stamps Allie and Loren need to buy stamps. Allie needs four $\text{\$0.49}$ stamps and nine $\text{\$0.02}$ stamps. Loren needs eight $\text{\$0.49}$ stamps and three $\text{\$0.02}$ stamps.

ⓐ How much will Allie’s stamps cost?

ⓑ How much will Loren’s stamps cost?

ⓒ What is the total cost of the girls’ stamps?

ⓓ How many $\text{\$0.49}$ stamps do the girls need altogether? How much will they cost?

ⓔ How many $\text{\$0.02}$ stamps do the girls need altogether? How much will they cost?
Counting Cash Grant is totaling up the cash from a fundraising dinner. In one envelope, he has twentythree $\text{\$5}$ bills, eighteen $\text{\$10}$ bills, and thirtyfour $\text{\$20}$ bills. In another envelope, he has fourteen $\text{\$5}$ bills, nine $\text{\$10}$ bills, and twentyseven $\text{\$20}$ bills.

ⓐ How much money is in the first envelope?

ⓑ How much money is in the second envelope?

ⓒ What is the total value of all the cash?

ⓓ What is the value of all the $\text{\$5}$ bills?

ⓔ What is the value of all $\text{\$10}$ bills?

ⓕ What is the value of all $\text{\$20}$ bills?
Writing Exercises
In your own words, state the Commutative Property of Addition and explain why it is useful.
In your own words, state the Associative Property of Multiplication and explain why it is useful.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?