7.2 Commutative and Associative Properties

Learning objectives

Learning objectives

• Use the commutative and associative properties

• Evaluate expressions using the commutative and associative properties

• Simplify expressions using the commutative and associative properties

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! $5\cdot{}3=3\cdot{}5.$ The order in which we multiply does not matter.

These examples illustrate the commutative properties of addition and multiplication.

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

Solution

1. Identify the operation in part ⓐ. The expression $-1+3$ contains addition, so the commutative property of addition allows the two addends to trade places.
2. Rewrite part ⓐ. Reversing the order gives $-1+3=3+(-1)$, and both orders evaluate to $2$.
3. Identify the operation in part ⓑ. The expression $4\cdot9$ contains multiplication, so the commutative property of multiplication allows the factors to trade places.
4. Rewrite part ⓑ. Reversing the order gives $4\cdot9=9\cdot4$, and both products equal $36$.

Takeaway: The commutative property permits reversing the order of addition or multiplication, so $-1+3=3+(-1)$ and $4(9)=9(4)$.

What about subtraction? Does order matter when we subtract numbers? Does $7-3$ give the same result as $3-7?$

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 $7+8\,\mathrm{is}\,15$ and then $15+2\,\mathrm{is}\,17.$ Others might start with $8+2\,\mathrm{makes}\,10$ and then $7+10\,\mathrm{makes}\,17.$

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.

Example 7.6

Solution

1. Recall the associative pattern. Associativity changes grouping but not order: $(a+b)+c=a+(b+c)$ and $(a\cdot{}b)\cdot{}c=a\cdot(b\cdot{}c)$.
2. Regroup the addition in part ⓐ. Change $(3+0.6)+0.4$ to $3+(0.6+0.4)$; since $0.6+0.4=1$, the expression becomes $3+1=4$.
3. Regroup the multiplication in part ⓑ. Change $\left(-4\cdot\frac{2}{5}\right)\cdot15$ to $-4\cdot\left(\frac{2}{5}\cdot15\right)$; since $\frac{2}{5}\cdot15=6$, the product is $-4\cdot6=-24$.

Takeaway: The associative property leaves the value unchanged while regrouping, so $0.6$ with $0.4$ and $\rac{2}{5}$ with $15$ make the arithmetic direct.

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

Example 7.7

Solution

1. Choose the property to apply. The product $6(3x)$ has three factors, $6$, $3$, and $x$, so the Associative Property of Multiplication permits regrouping the two known numbers.
2. Regroup the numerical factors. Rewrite $6(3x)$ as $(6\cdot3)x$, which groups the two numbers together and carries $x$ along as a factor.
3. Multiply the grouped numbers. Since $6\cdot3=18$, the expression becomes $18x$; no value for $x$ is needed.

Takeaway: Regrouping a product multiplies the known factors first, so $6(3x)$ simplifies directly to $18x$.

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.

Example 7.8

Solution

1. Substitute the value of $x$. With $x=\rac{7}{8}=0.875$, part ⓐ becomes $0.875+0.37+(-0.875)$ and part ⓑ becomes $0.875+(-0.875)+0.37$.
2. Regroup the opposite terms. In both parts, $0.875$ and $-0.875$ are additive inverses, so their sum is $0$. This leaves the remaining term $0.37$ unchanged in each expression.
3. Check the illustrated rearrangements. The figures show the same cancellation first with the symbolic terms $x+0.37+(-x)$ and $x+(-x)+0.37$, then with their decimal and fractional values, and they display the surviving value $0.37$ after the opposite pair is removed.
4. State both results. Part ⓐ equals $0.37$, and part ⓑ also equals $0.37$, confirming that changing the order did not change the value.

Takeaway: Additive inverses cancel to zero, so rearrange a sum to place an opposite pair together before doing any other arithmetic.

Let’s do one more, this time with multiplication.

Example 7.9

Solution

1. Use the equivalence of the two groupings. By the Associative Property of Multiplication, $\frac{4}{3}\left(\frac{3}{4}n\right)=\left(\frac{4}{3}\cdot\frac{3}{4}\right)n$, so the grouping cannot change the result.
2. Evaluate part ⓐ with its original grouping. Substitute $n=17$, giving $\frac{4}{3}\left(\frac{3}{4}\cdot17\right)$; since $\frac{3}{4}\cdot17=\frac{51}{4}$, the product is $\frac{4}{3}\cdot\frac{51}{4}=17$.
3. Evaluate part ⓑ with the new grouping. The factors $\frac{4}{3}$ and $\frac{3}{4}$ are reciprocals, so their product is $1$ and the expression becomes $1\cdot17$.
4. Compare the two results. Both part ⓐ and part ⓑ equal $17$, confirming that only the grouping changed.

Takeaway: Both parts equal $17$ because the associative and commutative properties let you regroup the reciprocal factors $\rac{4}{3}$ and $\rac{3}{4}$ into the product $1$.

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

Solution

1. Look for opposite terms. The terms $-84n$ and $84n$ are opposites because their coefficients sum to zero.
2. Reorder with the commutative property. Rewrite $-84n+(-73n)+84n$ as $-84n+84n+(-73n)$, placing the opposite pair next to each other.
3. Cancel the additive inverses. The sum $-84n+84n$ equals $0$, leaving $0+(-73n)$.
4. Simplify the remaining sum. Adding $0$ leaves $-73n$ unchanged, so the expression is $-73n$.

Takeaway: Reordering additive terms to place opposites together cancels $-84n$ with $84n$, leaving the simplified expression $-73n$.

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is $1.$

Example 7.11

Solution

1. Recognize reciprocal factors. The factors $\frac{7}{15}$ and $\frac{15}{7}$ are reciprocals because $\frac{7}{15}\cdot\frac{15}{7}=\frac{7\cdot15}{15\cdot7}=1$.
2. Reorder the factors. Use the Commutative Property of Multiplication to write $\frac{7}{15}\cdot\frac{8}{23}\cdot\frac{15}{7}$ as $\frac{7}{15}\cdot\frac{15}{7}\cdot\frac{8}{23}$.
3. Multiply the reciprocal pair first. The product $\frac{7}{15}\cdot\frac{15}{7}$ is $1$, so the expression becomes $1\cdot\frac{8}{23}$.
4. Apply the multiplicative identity. Since multiplying by $1$ changes nothing, the simplified result is $\frac{8}{23}$.

Takeaway: Pairing reciprocal factors $\rac{7}{15}$ and $\rac{15}{7}$ produces $1$, so the product simplifies to $\rac{8}{23}$.

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

Example 7.12

Solution

1. Find a common-denominator pair. The terms $\frac{3}{4}$ and $\frac{1}{4}$ have the same denominator, so their sum is immediate.
2. Regroup the sum. Use the associative property to rewrite $\left(\frac{5}{13}+\frac{3}{4}\right)+\frac{1}{4}$ as $\frac{5}{13}+\left(\frac{3}{4}+\frac{1}{4}\right)$.
3. Add inside the parentheses. Since $\frac{3}{4}+\frac{1}{4}=\frac{4}{4}=1$, the expression becomes $\frac{5}{13}+1$.
4. Write the sum with one denominator. Because $1=\frac{13}{13}$, the sum is $\frac{5}{13}+\frac{13}{13}=\frac{18}{13}$.

Takeaway: Regrouping $\rac{3}{4}$ with $\rac{1}{4}$ gives $1$, and adding it to $\rac{5}{13}$ yields the improper fraction $\rac{18}{13}$.

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

Example 7.13

Solution

1. Look for a whole-number sum. The coefficients $9.99$ and $1.01$ sum to $11$, a whole number, so grouping them first avoids adding several decimal places.
2. Regroup the terms. Rewrite $(6.47q+9.99q)+1.01q$ as $6.47q+(9.99q+1.01q)$.
3. Add within the parentheses. Since $9.99+1.01=11.00$, the grouped term is $11q$, leaving $6.47q+11q$.
4. Combine the coefficients. Adding $6.47$ and $11$ gives $17.47$, so the simplified expression is $17.47q$.

Takeaway: Grouping $9.99$ with $1.01$ makes the decimal sum $11$, so $(6.47q+9.99q)+1.01q$ simplifies to $17.47q$.

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

Solution

1. Look for a simpler product. The factors $8$ and $0.25$ multiply to the whole number $2$, so that pair is easier to evaluate first.
2. Regroup the factors. Use the associative property to rewrite $[1.67(8)](0.25)$ as $1.67[(8)(0.25)]$.
3. Multiply the grouped factors. Since $8\cdot0.25=2$, the expression becomes $1.67\cdot2$.
4. Complete the multiplication. Because $1.67\cdot2=3.34$, the simplified value is $3.34$.

Takeaway: Regrouping to multiply $8(0.25)=2$ first reduces $[1.67(8)](0.25)$ to the product $3.34$.

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.

Example 7.15

Solution

1. Separate the known numbers from the variable. In $6(9x)$, the factors are $6$, $9$, and $x$; the two numbers can be combined first.
2. Regroup with the associative property. Rewrite $6(9x)$ as $(6\cdot9)x$, which changes only how the factors are grouped.
3. Multiply the known factors. Since $6\cdot9=54$, the expression becomes $54x$.

Takeaway: By the associative property, $6(9x)$ can be regrouped as $(6\cdot{}9)x$, which gives $54x$.

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

Solution

1. Group like terms. Use the commutative property of addition to reorder $18p+6q+(-15p)+5q$ as $18p+(-15p)+6q+5q$.
2. Combine the $p$-terms. Since $18p+(-15p)=(18-15)p$, the $p$-part combines to $3p$.
3. Combine the $q$-terms. Since $6q+5q=(6+5)q$, the $q$-part combines to $11q$.
4. State the simplified expression. Adding the two variable parts gives $3p+11q$.

Takeaway: Rearrange the terms so like terms are adjacent, then combine $18p+(-15p)$ and $6q+5q$ to obtain $3p+11q$.

Section 7.2 Exercises

Section 7.2 Exercises

Use the Commutative and Associative Properties

In the following exercises, use the commutative properties to rewrite the given expression.

In the following exercises, use the associative properties to rewrite the given expression.

Evaluate Expressions using the Commutative and Associative Properties

In the following exercises, evaluate each expression for the given value.

Simplify Expressions Using the Commutative and Associative Properties

In the following exercises, simplify.

Everyday Math

Writing Exercises

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?

1. Use the commutative properties to rewrite the following:

1. ⓐ $\,-4+7=\mathrm{\_\_\_\_\_}\,$

2. ⓑ $\,6\cdot{}12=\mathrm{\_\_\_\_\_}$

2. Use the commutative properties to rewrite the following:

1. ⓐ $\,14+(-2)=\mathrm{\_\_\_\_\_}\,$

2. ⓑ $\,3(-5)=\mathrm{\_\_\_\_\_}$

3. Use the associative properties to rewrite the following:ⓐ $(1+0.7)+0.3=\mathrm{\_\_\_\_\_\_\_\_\_\_}\,$ ⓑ $\,(-9\cdot{}8)\cdot\frac{3}{4}=\mathrm{\_\_\_\_\_\_\_\_\_\_}$

4. Use the associative properties to rewrite the following:ⓐ $(4+0.6)+0.4=\mathrm{\_\_\_\_\_\_\_\_\_\_}\,$ ⓑ $\,(-2\cdot{}12)\cdot\frac{5}{6}=\mathrm{\_\_\_\_\_\_\_\_\_\_}$

1. Use the Associative Property of Multiplication to simplify the given expression: $8(4x).$

2. Use the Associative Property of Multiplication to simplify the given expression: $-9(7y).$

3. Evaluate each expression when $y=\frac{3}{8}:\,$ⓐ $\,y+0.84+(-y)$ⓑ $\,y+(-y)+0.84.$

4. Evaluate each expression when $f=\frac{17}{20}:\,$ⓐ $\,f+0.975+(-f)$ⓑ $\,f+(-f)+0.975.$

1. Evaluate each expression when $p=24:\,$ⓐ $\,\frac{5}{9}(\frac{9}{5}p)\,$ ⓑ $\,(\frac{5}{9}\cdot\frac{9}{5})p.$

2. Evaluate each expression when $q=15:\,$ⓐ $\,\frac{7}{11}(\frac{11}{7}q)\,$ ⓑ $\,(\frac{7}{11}\cdot\frac{11}{7})q$

3. Simplify: $-27a+(-48a)+27a.$

4. Simplify: $39x+(-92x)+(-39x).$

1. Simplify: $\frac{9}{16}\cdot\frac{5}{49}\cdot\frac{16}{9}.$

2. Simplify: $\frac{6}{17}\cdot\frac{11}{25}\cdot\frac{17}{6}.$

3. Simplify: $(\frac{7}{15}+\frac{5}{8})+\frac{3}{8}.$

4. Simplify: $(\frac{2}{9}+\frac{7}{12})+\frac{5}{12}.$

1. Simplify: $(5.58c+8.75c)+1.25c.$

2. Simplify: $(8.79d+3.55d)+5.45d.$

3. Simplify: $8(3y).$

4. Simplify: $12(5z).$

1. Simplify: $23r+14s+9r+(-15s).$

2. Simplify: $37m+21n+4m+(-15n).$

3. $7+6=\mathrm{\_\_\_}$

4. $7(-13)=\mathrm{\_\_\_}$

1. $(-12)(-18)=\mathrm{\_\_\_}$

2. $-15+7=\mathrm{\_\_\_}$

3. $y+1=\mathrm{\_\_\_}$

4. $-3m=\mathrm{\_\_\_}$

1. $(21+14)+9=\mathrm{\_\_\_}$

2. $(14\cdot{}6)\cdot{}9=\mathrm{\_\_\_}$

3. $(-2+6)+7=\mathrm{\_\_\_}$

4. $(13\cdot\frac{2}{3})\cdot{}18=\mathrm{\_\_\_}$

1. $4(7x)=\mathrm{\_\_\_}$

2. $(17+y)+33=\mathrm{\_\_\_}$

3. If $z=\frac{7}{8},$ evaluate:

• ⓐ $\,z+0.97+(-z)$

• ⓑ $\,z+(-z)+0.97$

4. If $d=-\frac{9}{4},$ evaluate:

1. ⓐ $\,d+2.375+(-d)$

2. ⓑ $\,d+(-d)+2.375$

1. If $k=21,$ evaluate:

1. ⓐ $\,\frac{4}{13}(\frac{13}{4}k)$

2. ⓑ $\,(\frac{4}{13}\cdot\frac{13}{4})k$

2. If $n=-8,$ evaluate:

1. ⓐ $\,-\frac{5}{21}(\frac{21}{5}n)$

2. ⓑ $\,(-\frac{5}{21}\cdot\frac{21}{5})n$

3. $\frac{2}{5}+\frac{5}{12}+(-\frac{2}{5})$

4. $\frac{13}{18}\cdot\frac{25}{7}\cdot\frac{18}{13}$

1. $\frac{3}{10}\cdot\frac{13}{23}\cdot\frac{50}{3}$

2. $-36\cdot{}11\cdot\frac{4}{9}$

3. $(\frac{1}{12}+\frac{4}{9})+\frac{5}{9}$

4. $\frac{8}{15}+\frac{5}{7}+\frac{2}{7}$

1. $(5.89d+2.75d)+1.25d$

2. $9(8w)$

3. $-23(2n)$

4. $20(\frac{3}{5}q)$

1. $15u+11v+27u+19v$

2. $-22p+17q+(-35p)+(-27q)$

3. $\frac{5}{6}a+\frac{3}{10}b+\frac{1}{6}a+\frac{9}{10}b$

4. $9.6m+7.22n+(-2.19m)+(-0.65n)$

1. Counting Cash Grant is totaling up the cash from a fundraising dinner. In one envelope, he has twenty-three $\text{\textdollar 5}$ bills, eighteen $\text{\textdollar 10}$ bills, and thirty-four $\text{\textdollar 20}$ bills. In another envelope, he has fourteen $\text{\textdollar 5}$ bills, nine $\text{\textdollar 10}$ bills, and twenty-seven $\text{\textdollar 20}$ bills.

1. ⓐ How much money is in the first envelope?

2. ⓑ How much money is in the second envelope?

3. ⓒ What is the total value of all the cash?

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

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

6. ⓕ What is the value of all $\text{\textdollar 20}$ bills?

2. In your own words, state the Associative Property of Multiplication and explain why it is useful.