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By the end of this section, you will be able to:
  • Use the commutative and associative properties
  • Evaluate expressions using the commutative and associative properties
  • Simplify expressions using the commutative and associative properties

Before you get started, take this readiness quiz.

  1. Simplify: 7 y + 2 + y + 13 .
    If you missed this problem, review Evaluate, Simplify and Translate Expressions .
  2. Multiply: 2 3 · 18 .
    If you missed this problem, review Multiply and Divide Fractions .
  3. Find the opposite of 15 .
    If you missed this problem, review Subtract Integers .

In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations in the next chapter.

Use the commutative and associative properties

Think about adding two numbers, such as 5 and 3 .

5 + 3 3 + 5 8 8

The results are the same. 5 + 3 = 3 + 5

Notice, the order in which we add does not matter. The same is true when multiplying 5 and 3 .

5 · 3 3 · 5 15 15

Again, the results are the same! 5 · 3 = 3 · 5 . The order in which we multiply does not matter.

These examples illustrate the commutative properties of addition and multiplication.

Commutative properties

Commutative Property of Addition : if a and b are real numbers, then

a + b = b + a

Commutative Property of Multiplication : if a and b are real numbers, then

a · b = b · a

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

Use the commutative properties to rewrite the following expressions:

−1 + 3 = _____

4 · 9 = _____

Solution

−1 + 3 = _____
Use the commutative property of addition to change the order. −1 + 3 = 3 + ( −1 )
4 · 9 = _____
Use the commutative property of multiplication to change the order. 4 · 9 = 9 · 4
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the commutative properties to rewrite the following:

  1. −4 + 7 = _____
  2. 6 · 12 = _____

  1. −4 + 7 = 7 + (−4)
  2. 6 · 12 = 12 · 6
Got questions? Get instant answers now!

Use the commutative properties to rewrite the following:

  1. 14 + ( −2 ) = _____
  2. 3 ( −5 ) = _____

  1. 14 + (−2) = −2 + 14
  2. 3(−5) = (−5)3
Got questions? Get instant answers now!

What about subtraction? Does order matter when we subtract numbers? Does 7 3 give the same result as 3 7 ?

7 3 3 7 4 4 4 −4
The results are not the same. 7 3 3 7

Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.

Let’s see what happens when we divide two numbers. Is division commutative?

12 ÷ 4 4 ÷ 12 12 4 4 12 3 1 3 3 1 3
The results are not the same. So 12 ÷ 4 4 ÷ 12

Since changing the order of the division did not give the same result, division is not commutative.

Addition and multiplication are commutative. Subtraction and division are not commutative.

Suppose you were asked to simplify this expression.

7 + 8 + 2

How would you do it and what would your answer be?

Some people would think 7 + 8 is 15 and then 15 + 2 is 17 . Others might start with 8 + 2 makes 10 and then 7 + 10 makes 17 .

Both ways give the same result, as shown in [link] . (Remember that parentheses are grouping symbols that indicate which operations should be done first.)

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
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?
Chris Reply

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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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