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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 = _____


−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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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
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
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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