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Simplify: ( x + 2 ) p .

xp + 2 p

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Simplify: ( y + 4 ) q .

yq + 4 q

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When you distribute a negative number, you need to be extra careful to get the signs correct.

Simplify: −2 ( 4 y + 1 ) .

Solution

.
Distribute. .
Simplify. .
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Simplify: −3 ( 6 m + 5 ) .

−18 m − 15

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Simplify: −6 ( 8 n + 11 ) .

−48 n − 66

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Simplify: −11 ( 4 3 a ) .

Solution

.
Distribute. .
Multiply. .
Simplify. .

You could also write the result as 33 a 44 . Do you know why?

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Simplify: −5 ( 2 3 a ) .

−10 + 15 a

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Simplify: −7 ( 8 15 y ) .

−56 + 105 y

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In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, a = −1 · a .

Simplify: ( y + 5 ) .

Solution

.
Multiplying by -1 results in the opposite. .
Distribute. .
Simplify. .
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Simplify: ( z 11 ) .

z + 11

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Simplify: ( x 4 ) .

x + 4

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Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Simplify: 8 2 ( x + 3 ) .

Solution

.
Distribute. .
Multiply. .
Combine like terms. .
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Simplify: 9 3 ( x + 2 ) .

−3 x + 3

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Simplify: 7 x 5 ( x + 4 ) .

2 x − 20

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Simplify: 4 ( x 8 ) ( x + 3 ) .

Solution

.
Distribute. .
Combine like terms. .
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Simplify: 6 ( x 9 ) ( x + 12 ) .

5 x − 66

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Simplify: 8 ( x 1 ) ( x + 5 ) .

7 x − 13

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Evaluate expressions using the distributive property

Some students need to be convinced that the Distributive Property always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part , we will evaluate the form with parentheses, and in part we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

When y = 10 evaluate: 6 ( 5 y + 1 ) 6 · 5 y + 6 · 1 .

Solution

6 ( 5 y + 1 )
. .
Simplify in the parentheses. 6 ( 51 )
Multiply. 306
6 · 5 y + 6 · 1
. .
Simplify in the parentheses. .
Add. .

Notice, the answers are the same. When y = 10 ,

6 ( 5 y + 1 ) = 6 · 5 y + 6 · 1 .

Try it yourself for a different value of y .

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Evaluate when w = 3 : 5 ( 5 w + 9 ) 5 · 5 w + 5 · 9 .

  1. 120
  2. 120

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Evaluate when y = 2 : 9 ( 3 y + 8 ) 9 · 3 y + 9 · 8 .

  1. 126
  2. 126

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When y = 3 , evaluate −2 ( 4 y + 1 ) −2 · 4 y + ( −2 ) · 1 .

Solution

−2 ( 4 y + 1 )
. .
Simplify in the parentheses. −2 ( 13 )
Multiply. −26
−2 · 4 y + ( −2 ) · 1
. .
Multiply. −24 2
Subtract. −26
The answers are the same. When y = 3 , −2 ( 4 y + 1 ) = −8 y 2
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Evaluate when n = −2 : −6 ( 8 n + 11 ) −6 · 8 n + ( −6 ) · 11 .

  1. 30
  2. 30

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Evaluate when m = −1 : −3 ( 6 m + 5 ) −3 · 6 m + ( −3 ) · 5 .

  1. 3
  2. 3

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When y = 35 evaluate ( y + 5 ) and y 5 to show that ( y + 5 ) = y 5 .

Solution

( y + 5 )
. .
Add in the parentheses. - ( 40 )
Simplify. −40
y 5
. .
Simplify. −40
The answers are the same when y = 35 , demonstrating that ( y + 5 ) = y 5
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Evaluate when x = 36 : ( x 4 ) x + 4 to show that ( x 4 ) = x 4 .

  1. −32
  2. −32

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Evaluate when z = 55 : ( z 10 ) z + 10 to show that ( z 10 ) = z + 10 .

  1. −45
  2. −45

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Access additional online resources

Key concepts

  • Distributive Property:
    • If a , b , c are real numbers then
      • a ( b + c ) = a b + a c
      • ( b + c ) a = b a + c a
      • a ( b c ) = a b a c

Practice makes perfect

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

7 ( y 13 )

7 y − 91

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5 ( 7 u 4 )

35 u − 20

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1 5 ( 4 m + 20 )

4 5 m + 4

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10 ( 3 10 x 2 5 )

3 x − 4

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12 ( 1 6 + 3 4 s )

2 + 9 s

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u ( v 10 )

uv − 10 u

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−3 ( a + 11 )

−3 a − 33

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−9 ( 9 a + 4 )

−81 a − 36

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−4 ( q 7 )

−4 q + 28

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−6 ( 7 x 8 )

−42 x + 48

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( q + 11 )

q − 11

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( 5 p 4 )

−5 p + 4

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12 + 8 ( u 1 )

8 u + 4

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18 4 ( x + 2 )

−4 x + 10

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9 6 ( 7 n 5 )

−42 n + 39

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8 ( r 7 )

r + 15

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−4 ( c 10 )

c + 6

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( 5 m 3 ) ( m + 7 )

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( 4 y 1 ) ( y 2 )

3 y + 1

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5 ( 2 n + 9 ) + 12 ( n 3 )

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9 ( 5 u + 8 ) + 2 ( u 6 )

47 u + 60

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9 ( 8 x 3 ) ( −2 )

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4 ( 6 x 1 ) ( −8 )

24 x + 4

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14 ( c 1 ) 8 ( c 6 )

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11 ( n 7 ) 5 ( n 1 )

6 n − 72

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6 ( 7 y + 8 ) ( 30 y 15 )

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7 ( 3 n + 9 ) ( 4 n 13 )

17 n + 76

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Evaluate Expressions Using the Distributive Property

In the following exercises, evaluate both expressions for the given value.

If v = −2 , evaluate

  1. 6 ( 4 v + 7 )
  2. 6 · 4 v + 6 · 7

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If u = −1 , evaluate

  1. 8 ( 5 u + 12 )
  2. 8 · 5 u + 8 · 12

  1. 56
  2. 56

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If n = 2 3 , evaluate

  1. 3 ( n + 5 6 )
  2. 3 · n + 3 · 5 6

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If y = 3 4 , evaluate

  1. 4 ( y + 3 8 )
  2. 4 · y + 4 · 3 8

  1. 9 2
  2. 9 2

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If y = 7 12 , evaluate

  1. −3 ( 4 y + 15 )
  2. 3 · 4 y + ( −3 ) · 15

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If p = 23 30 , evaluate

  1. −6 ( 5 p + 11 )
  2. −6 · 5 p + ( −6 ) · 11

  1. −89
  2. −89

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If m = 0.4 , evaluate

  1. −10 ( 3 m 0.9 )
  2. −10 · 3 m ( −10 ) ( 0.9 )

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If n = 0.75 , evaluate

  1. −100 ( 5 n + 1.5 )
  2. −100 · 5 n + ( −100 ) ( 1.5 )

  1. −525
  2. −525

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If y = −25 , evaluate

  1. ( y 25 )
  2. y + 25

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If w = −80 , evaluate

  1. ( w 80 )
  2. w + 80

  1. 160
  2. 160

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If p = 0.19 , evaluate

  1. ( p + 0.72 )
  2. p 0.72

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If q = 0.55 , evaluate

  1. ( q + 0.48 )
  2. q 0.48

  1. −1.03
  2. −1.03

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Everyday math

Buying by the case Joe can buy his favorite ice tea at a convenience store for $1.99 per bottle. At the grocery store, he can buy a case of 12 bottles for $23.88 .

Use the distributive property to find the cost of 12 bottles bought individually at the convenience store. (Hint: notice that $1.99 is $2 $0.01 . )

Is it a bargain to buy the iced tea at the grocery store by the case?

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Multi-pack purchase Adele’s shampoo sells for $3.97 per bottle at the drug store. At the warehouse store, the same shampoo is sold as a 3-pack for $10.49 .

Show how you can use the distributive property to find the cost of 3 bottles bought individually at the drug store.

How much would Adele save by buying the 3-pack at the warehouse store?

  1. 3(4 − 0.03) = 11.91
  2. $1.42

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Writing exercises

Simplify 8 ( x 1 4 ) using the distributive property and explain each step.

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Explain how you can multiply 4 ( $5.97 ) without paper or a calculator by thinking of $5.97 as 6 0.03 and then using the distributive property.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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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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