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Simplify the expression: [ 1.67 ( 8 ) ] ( 0.25 ) .

Solution

Notice that multiplying ( 8 ) ( 0.25 ) is easier than multiplying 1.67 ( 8 ) because it gives a whole number. (Think about having 8 quarters—that makes $2.)

[ 1.67 ( 8 ) ] ( 0.25 )
Regroup. 1.67 [ ( 8 ) ( 0.25 ) ]
Multiply in the brackets first. 1.67 [ 2 ]
Multiply. 3.34
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Simplify: [ 1.17 ( 4 ) ] ( 2.25 ) .

10.53

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Simplify: [ 3.52 ( 8 ) ] ( 2.5 ) .

70.4

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When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.

Simplify: 6 ( 9 x ) .

Solution

6 ( 9 x )
Use the associative property of multiplication to re-group. ( 6 · 9 ) x
Multiply in the parentheses. 54 x
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Simplify: 8 ( 3 y ) .

24 y

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Simplify: 12 ( 5 z ) .

60 z

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In The Language of Algebra , we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression 3 x + 7 + 4 x + 5 by rewriting it as 3 x + 4 x + 7 + 5 and then simplified it to 7 x + 12 . We were using the Commutative Property of Addition.

Simplify: 18 p + 6 q + ( −15 p ) + 5 q .

Solution

Use the Commutative Property of Addition to re-order so that like terms are together.

18 p + 6 q + ( −15 p ) + 5 q
Re-order terms. 18 p + ( −15 p ) + 6 q + 5 q
Combine like terms. 3 p + 11 q
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Simplify: 23 r + 14 s + 9 r + ( −15 s ) .

32 r s

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Simplify: 37 m + 21 n + 4 m + ( −15 n ) .

41 m + 6 n

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Key concepts

  • Commutative Properties
    • Commutative Property of Addition:
      • If a , b are real numbers, then a + b = b + a
    • Commutative Property of Multiplication:
      • If a , b are real numbers, then a b = b a
  • Associative Properties
    • Associative Property of Addition:
      • If a , b , c are real numbers then ( a + b ) + c = a + ( b + c )
    • Associative Property of Multiplication:
      • If a , b , c are real numbers then ( a b ) c = a ( b c )

Practice makes perfect

Use the Commutative and Associative Properties

In the following exercises, use the commutative properties to rewrite the given expression.

7 + 6 = ___

7 + 6 = 6 + 7

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7 ( −13 ) = ___

7(−13) = (−13)7

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( −19 ) ( −14 ) = ___

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( −12 ) ( −18 ) = ___

(−12)(−18) = (−18)(−12)

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−15 + 7 = ___

−15 + 7 = 7 + (−15)

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y + 1 = ___

y + 1 = 1 + y

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−3 m = ___

−3 m = m (−3)

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In the following exercises, use the associative properties to rewrite the given expression.

( 21 + 14 ) + 9 = ___

(21 + 14) + 9 = 21 + (14 + 9)

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( 14 · 6 ) · 9 = ___

(14 · 6) · 9 = 14(6 · 9)

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( −2 + 6 ) + 7 = ___

(−2 + 6) + 7 = −2 + (6 + 7)

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( 16 · 4 5 ) · 15 = ___

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( 13 · 2 3 ) · 18 = ___

( 13 · 2 3 ) · 18 = 13 ( 2 3 · 18 )

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4 ( 7 x ) = ___

4(7 x ) = (4 · 7) x

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( 17 + y ) + 33 = ___

(17 + y ) + 33 = 17 + ( y + 33)

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Evaluate Expressions using the Commutative and Associative Properties

In the following exercises, evaluate each expression for the given value.

If y = 5 8 , evaluate:

  1. y + 0.49 + ( y )
  2. y + ( y ) + 0.49

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If z = 7 8 , evaluate:

  1. z + 0.97 + ( z )
  2. z + ( z ) + 0.97

  1. 0.97
  2. 0.97
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If c = 11 4 , evaluate:

  1. c + 3.125 + ( c )
  2. c + ( c ) + 3.125

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If d = 9 4 , evaluate:

  1. d + 2.375 + ( d )
  2. d + ( d ) + 2.375

  1. 2.375
  2. 2.375

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If j = 11 , evaluate:

  1. 5 6 ( 6 5 j )
  2. ( 5 6 · 6 5 ) j

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If k = 21 , evaluate:

  1. 4 13 ( 13 4 k )
  2. ( 4 13 · 13 4 ) k

  1. 21
  2. 21

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If m = −25 , evaluate:

  1. 3 7 ( 7 3 m )
  2. ( 3 7 · 7 3 ) m

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If n = −8 , evaluate:

  1. 5 21 ( 21 5 n )
  2. ( 5 21 · 21 5 ) n

  1. −8
  2. −8

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Simplify Expressions Using the Commutative and Associative Properties

In the following exercises, simplify.

9 y + 23 + ( −9 y )

23

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2 5 + 5 12 + ( 2 5 )

5 12

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13 18 · 25 7 · 18 13


25 7

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3 10 · 13 23 · 50 3

65 23

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−36 · 11 · 4 9

−176

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( 1 12 + 4 9 ) + 5 9

13 12

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( 4.33 p + 1.09 p ) + 3.91 p

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( 5.89 d + 2.75 d ) + 1.25 d

9.89 d

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[ 9.731 ( 4 ) ] ( 0.75 )

29.193

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14 x + 19 y + 25 x + 3 y

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15 u + 11 v + 27 u + 19 v

42 u + 30 v

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43 m + ( −12 n ) + ( −16 m ) + ( −9 n )

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−22 p + 17 q + ( −35 p ) + ( −27 q )

−57 p + (−10 q )

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3 8 g + 1 12 h + 7 8 g + 5 12 h

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5 6 a + 3 10 b + 1 6 a + 9 10 b

a + 6 5 b

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6.8 p + 9.14 q + ( −4.37 p ) + ( −0.88 q )

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9.6 m + 7.22 n + ( −2.19 m ) + ( −0.65 n )

7.41 m + 6.57 n

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Everyday math

Stamps Allie and Loren need to buy stamps. Allie needs four $0.49 stamps and nine $0.02 stamps. Loren needs eight $0.49 stamps and three $0.02 stamps.

How much will Allie’s stamps cost?

How much will Loren’s stamps cost?

What is the total cost of the girls’ stamps?

How many $0.49 stamps do the girls need altogether? How much will they cost?

How many $0.02 stamps do the girls need altogether? How much will they cost?

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Counting Cash Grant is totaling up the cash from a fundraising dinner. In one envelope, he has twenty-three $5 bills, eighteen $10 bills, and thirty-four $20 bills. In another envelope, he has fourteen $5 bills, nine $10 bills, and twenty-seven $20 bills.

How much money is in the first envelope?

How much money is in the second envelope?

What is the total value of all the cash?

What is the value of all the $5 bills?

What is the value of all $10 bills?

What is the value of all $20 bills?

  1. $975
  2. $700
  3. $1675
  4. $185
  5. $270
  6. $1220
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Writing exercises

In your own words, state the Commutative Property of Addition and explain why it is useful.

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In your own words, state the Associative Property of Multiplication and explain why it is useful.

Answers will vary.

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Self check

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

.

After reviewing this checklist, what will you do to become confident for all objectives?

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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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.
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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.
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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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