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The associative property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.

We saw that subtraction and division were not commutative. They are not associative either.

When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the commutative property of addition to write the like terms together.

Simplify: 18 p + 6 q + 15 p + 5 q .

Solution

18 p + 6 q + 15 p + 5 q Use the commutative property of addition to re-order so that like terms are together. 18 p + 15 p + 6 q + 5 q Add like terms. 33 p + 11 q

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Simplify: 23 r + 14 s + 9 r + 15 s .

32 r + 29 s

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Simplify: 37 m + 21 n + 4 m 15 n .

41 m + 6 n

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When we have to simplify algebraic expression    s, we can often make the work easier by applying the commutative or associative property first, instead of automatically following the order of operations. When adding or subtracting fractions, combine those with a common denominator first.

Simplify: ( 5 13 + 3 4 ) + 1 4 .

Solution

( 5 13 + 3 4 ) + 1 4 Notice that the last 2 terms have a common denominator, so change the grouping. 5 13 + ( 3 4 + 1 4 ) Add in parentheses first. 5 13 + ( 4 4 ) Simplify the fraction. 5 13 + 1 Add. 1 5 13 Convert to an improper fraction. 18 13

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Simplify: ( 7 15 + 5 8 ) + 3 8 .

1 7 15

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Simplify: ( 2 9 + 7 12 ) + 5 12 .

1 2 9

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Use the associative property to simplify 6 ( 3 x ) .

Solution

Use the associative property of multiplication, ( a · b ) · c = a · ( b · c ) , to change the grouping.

6 ( 3 x ) Change the grouping. ( 6 · 3 ) x Multiply in the parentheses. 18 x

Notice that we can multiply 6 · 3 but we could not multiply 3 x without having a value for x .

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Use the associative property to simplify 8(4 x ).

32 x

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Use the associative property to simplify −9 ( 7 y ) .

−63 y

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Use the identity and inverse properties of addition and multiplication

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity    .

For example,

13 + 0 −14 + 0 0 + ( −8 ) 13 14 8

These examples illustrate the Identity Property of Addition that states that for any real number a , a + 0 = a and 0 + a = a .

What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity    .

For example,

43 · 1 27 · 1 1 · 3 5 43 27 3 5

These examples illustrate the Identity Property of Multiplication that states that for any real number a , a · 1 = a and 1 · a = a .

We summarize the Identity Properties below.

Identity property

of addition For any real number a : a + 0 = a 0 + a = a 0 is the additive identity of multiplication For any real number a : a · 1 = a 1 · a = a 1 is the multiplicative identity
In the top line of this figure, we have the question “What number added to 5 gives the additive identity, 0?” On the following line, we have 5 plus a blank space equals 0. Then it is stated that “We know 5 plus negative 5 equals 0.” On the following line, we have the question “What number added to negative 6 gives the additive identity, 0?” On the following line, we have negative 6 plus a blank space equals 0. Then it is stated that “We know negative 6 plus 6 equals 0.”

Notice that in each case, the missing number was the opposite of the number!

We call a . the additive inverse    of a . The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number a , a + ( a ) = 0 . Remember, a number and its opposite add to zero.

What number multiplied by 2 3 gives the multiplicative identity, 1? In other words, 2 3 times what results in 1?

We have the statement that 2/3 times a blank space equals 1. Then it is stated that “We know 2/3 times 3/2 equals 1.”

What number multiplied by 2 gives the multiplicative identity, 1? In other words 2 times what results in 1?

Practice Key Terms 4

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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