7.2 Commutative and associative properties (Page 2/4)

Prealgebra Page 2 / 4

The image shows an equation. The left side of the equation shows the quantity 7 plus 8 in parentheses plus 2. The right side of the equation show 7 plus the quantity 8 plus 2. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 17 below. — 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:

5 \cdot \frac{1}{3} \cdot 3

Changing the grouping of the numbers gives the same result, as shown in [link] .

The image shows an equation. The left side of the equation shows the quantity 5 times 1 third in parentheses times 3. The right side of the equation show 5 times the quantity 1 third times 3. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 5 below. — 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

(a + b) + c = a + (b + c)

Associative Property of Multiplication : if $a, b,$ and $c$ are real numbers, then

(a \cdot b) \cdot c = a \cdot (b \cdot c)

Use the associative properties to rewrite the following:

ⓐ $(3 + 0.6) + 0.4 =__________$

ⓑ $(−4 \cdot \frac{2}{5}) \cdot 15 =__________$

Solution

ⓐ
	$(3 + 0.6) + 0.4 =__________$
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.

ⓑ
	$(−4 \cdot \frac{2}{5}) \cdot 15 =__________$
Change the grouping.	$(−4 \cdot \frac{2}{5}) \cdot 15 = −4 \cdot (\frac{2}{5} \cdot 15)$

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

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Use the associative properties to rewrite the following:
ⓐ $(1 + 0.7) + 0.3 =__________$ ⓑ $(−9 \cdot 8) \cdot \frac{3}{4} =__________$

ⓐ $(1 + 0.7) + 0.3 = 1 + (0.7 + 0.3)$
ⓑ $(−9 \cdot 8) \cdot \frac{3}{4} = −9 (8 \cdot \frac{3}{4})$

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Use the associative properties to rewrite the following:
ⓐ $(4 + 0.6) + 0.4 =__________$ ⓑ $(−2 \cdot 12) \cdot \frac{5}{6} =__________$

ⓐ $(4 + 0.6) + 0.4 = 4 + (0.6 + 0.4)$
ⓑ $(−2 \cdot 12) \cdot \frac{5}{6} = −2 (12 \cdot \frac{5}{6})$

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

Use the Associative Property of Multiplication to simplify: $6 (3 x) .$

Solution

	$6 (3 x)$
Change the grouping.	$(6 \cdot 3) x$
Multiply in the parentheses.	$18 x$

Notice that we can multiply $6 \cdot 3,$ but we could not multiply $3 \cdot x$ without having a value for $x .$

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

32 x

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

−63 y

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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.

Evaluate each expression when $x = \frac{7}{8} .$

ⓐ $x + 0.37 + (- x)$
ⓑ $x + (- x) + 0.37$

Solution

ⓐ

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?

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

ⓐ 0.84
ⓑ 0.84

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

ⓐ 0.975
ⓑ 0.975

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

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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