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By the end of this section, you will be able to:
  • Simplify expressions using the distributive property
  • Evaluate expressions using the distributive property

Before you get started, take this readiness quiz.

  1. Multiply: 3 ( 0.25 ) .
    If you missed this problem, review Decimal Operations
  2. Simplify: 10 ( −2 ) ( 3 ) .
    If you missed this problem, review Subtract Integers
  3. Combine like terms: 9 y + 17 + 3 y 2 .
    If you missed this problem, review Evaluate, Simplify and Translate Expressions .

Simplify expressions using the distributive property

Suppose three friends are going to the movies. They each need $9.25 ; that is, 9 dollars and 1 quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

The image shows the equation 3 times 9 equal to 27. Below the 3 is an image of three people. Below the 9 is an image of 9 one dollar bills. Below the 27 is an image of three groups of 9 one dollar bills for a total of 27 one dollar bills. The image shows the equation 3 times 25 cents equal to 75 cents. Below the 3 is an image of three people. Below the 25 cents is an image of a quarter. Below the 75 cents is an image of three quarters.

They need 3 times $9 , so $27 , and 3 times 1 quarter, so 75 cents. In total, they need $27.75 .

If you think about doing the math in this way, you are using the Distributive Property.

Distributive property

If a , b , c are real numbers, then

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

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

3 ( 9.25 ) 3 ( 9 + 0.25 ) 3 ( 9 ) + 3 ( 0.25 ) 27 + 0.75 27.75

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression 3 ( x + 4 ) , the order of operations says to work in the parentheses first. But we cannot add x and 4 , since they are not like terms. So we use the Distributive Property, as shown in [link] .

Simplify: 3 ( x + 4 ) .

Solution

3 ( x + 4 )
Distribute. 3 · x + 3 · 4
Multiply. 3 x + 12
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Simplify: 4 ( x + 2 ) .

4 x + 8

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Simplify: 6 ( x + 7 ) .

6 x + 42

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Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in [link] would look like this:

The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4. The image shows and equation. On the left side of the equation is the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4. This is set equal to 3 times x plus 3 times 4.

Simplify: 6 ( 5 y + 1 ) .

Solution

.
Distribute. .
Multiply. .
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Simplify: 9 ( 3 y + 8 ) .

27 y + 72

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Simplify: 5 ( 5 w + 9 ) .

25 w + 45

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The distributive property can be used to simplify expressions that look slightly different from a ( b + c ) . Here are two other forms.

Distributive property

If a , b , c are real numbers, then a ( b + c ) = a b + a c
Other forms: a ( b c ) = a b a c
( b + c ) a = b a + c a

Simplify: 2 ( x 3 ) .

Solution

.
Distribute. .
Multiply. .
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Simplify: 7 ( x 6 ) .

7 x − 42

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Simplify: 8 ( x 5 ) .

8 x − 40

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Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

Simplify: 3 4 ( n + 12 ) .

Solution

.
Distribute. .
Simplify. .
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Simplify: 2 5 ( p + 10 ) .

2 5 p + 4

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Simplify: 3 7 ( u + 21 ) .

3 7 u + 9

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Simplify: 8 ( 3 8 x + 1 4 ) .

Solution

.
Distribute. .
Multiply. .
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Simplify: 6 ( 5 6 y + 1 2 ) .

5 y + 3

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Simplify: 12 ( 1 3 n + 3 4 ) .

4 n + 9

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Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

Simplify: 100 ( 0.3 + 0.25 q ) .

Solution

.
Distribute. .
Multiply. .
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Simplify: 100 ( 0.7 + 0.15 p ) .

70 + 15 p

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Simplify: 100 ( 0.04 + 0.35 d ) .

4 + 35 d

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In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

Simplify: m ( n 4 ) .

Solution

.
Distribute. .
Multiply. .

Notice that we wrote m · 4 as 4 m . We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.

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Simplify: r ( s 2 ) .

rs − 2 r

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Simplify: y ( z 8 ) .

yz − 8 y

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The next example will use the ‘backwards’ form of the Distributive Property, ( b + c ) a = b a + c a .

Questions & Answers

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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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what is the actual application of fullerenes nowadays?
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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s. Reply
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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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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