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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
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How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
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Shanjida
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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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