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Add 3 and 5, then multiply this sum by 2.

  3 + 5 2 = 8 2 = 16 alignl { stack { size 12{`3+5 cdot 2} {} #size 12{`=8 cdot 2} {} # size 12{`="16"} {}} } {}

Multiply 5 and 2, then add 3 to this product.

  3 + 5 2 = 3 + 10 = 13 alignl { stack { size 12{`3+5 cdot 2} {} #size 12{`=3+"10"} {} # size 12{`="13"} {}} } {}

We now have two values for the same expression.

We need a set of rules to guide anyone to one unique value for this kind of expression. Some of these rules are based on convention, while other are forced on up by mathematical logic.

The universally agreed-upon accepted order of operations for evaluating a mathematical expression is as follows:

1. Parentheses (grouping symbols) from the inside out.

By parentheses we mean anything that acts as a grouping symbol, including anything inside symbols such as [  ], {  }, |  |, and size 12{` sqrt {`} } {} . Any expression in the numerator or denominator of a fraction or in an exponent is also considered grouped, and should be simplified before carrying out further operations.

If there are nested parentheses (parentheses inside parentheses), you work from the innermost parentheses outward.

2. Exponents and other special functions, such as log, sin, cos etc.

3. Multiplications and divisions, from left to right.

4. Additions and subtractions, from left to right.

For example, given: 3 + 15 ÷ 3 + 5 × 2 2+3

The exponent is an implied grouping, so the 2+3 must be evaluated first:

 = 3 + 15 ÷ 3 + 5 × 2 5

Now the exponent is carried out:

 = 3 +15 ÷ 3 + 5 × 32

Then the multiplication and division, left to right using 15 ÷ 3 = 5 and 5 × 32 = 160:

 = 3 + 5 + 160

Finally, the addition, left to right:

 = 168

Examples, order of operation

Determine the value of each of the following.

21 + 3 12 size 12{"21"+3 cdot "12"} {} .

Multiply first:

= 21 + 36 size 12{"21"+"36"} {}

Add.

= 57

  

( 15 8 ) + 5 ( 6 + 4 ) size 12{ \( "15" - 8 \) +5 \( 6+4 \) } {} .

Simplify inside parentheses first.

= 7 + 5 10 size 12{7+5 cdot "10"} {}

Multiply.

= 7 + 50 size 12{7+"50"} {}

Add.

= 57

  

63 ( 4 + 6 3 ) + 76 4 size 12{"63" - \( 4+6 cdot 3 \) +"76" - 4} {} .

Simplify first within the parentheses by multiplying, then adding:

= 63 ( 4 + 18 ) + 76 4 size 12{"63" - \( 4+"18" \) +"76" - 4} {}

= 63 22 + 76 4 size 12{"63" - "22"+"76" - 4} {}

Now perform the additions and subtractions, moving left to right:

= 41 + 76 4 size 12{"41"+"76" - 4} {}

= 117 4 size 12{"117" - 4} {}

= 113.

  

7 6 4 2 + 1 5 size 12{7 cdot 6 - 4 rSup { size 8{2} } +1 rSup { size 8{5} } } {}

Evaluate the exponential forms, moving from left to right:

= 7 6 16 + 1 size 12{7 cdot 6 - "16"+1} {}

Multiply 7 · 6:

= 42 16 + 1 size 12{"42" - "16"+1} {}

Subtract 16 from 42:

= 26 + 1

Add 26 and 1:

= 27.

  

6 ( 3 2 + 2 2 ) + 4 2 size 12{6 cdot \( 3 rSup { size 8{2} } +2 rSup { size 8{2} } \) +4 rSup { size 8{2} } } {}

Evaluate the exponential forms in the parentheses:

= 6 ( 9 + 4 ) + 4 2 size 12{6 cdot \( 9+4 \) +4 rSup { size 8{2} } } {}

Add 9 and 4 in the parentheses:

= 6 ( 13 ) + 4 2 size 12{6 cdot \( "13" \) +4 rSup { size 8{2} } } {}

Evaluate the exponential form 4 2 size 12{4 rSup { size 8{2} } } {} :

= 6 ( 13 ) + 16 size 12{6 cdot \( "13" \) +"16"} {}

Multiply 6 and 13:

= 78 + 16 size 12{"78"+"16"} {}

Add 78 and 16:

= 94

  

6 2 + 2 2 4 2 + 6 2 2 + 1 3 + 8 2 10 2 19 5 size 12{ { {6 rSup { size 8{2} } +2 rSup { size 8{2} } } over {4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } } } + { {1 rSup { size 8{3} } +8 rSup { size 8{2} } } over {"10" rSup { size 8{2} } - "19" cdot 5} } } {} .

= 36 + 4 16 + 6 4 + 1 + 64 100 19 5 size 12{ { {"36"+4} over {"16"+6 cdot 4} } + { {1+"64"} over {"100" - "19" cdot 5} } } {}

= 36 + 4 16 + 24 + 1 + 64 100 95 size 12{ { {"36"+4} over {"16"+"24"} } + { {1+"64"} over {"100" - "95"} } } {}

= 40 40 + 65 5 size 12{ { {"40"} over {"40"} } + { {"65"} over {5} } } {}

= 1+13

= 14

Recall that the bar is a grouping symbol. The fraction 6 2 + 2 2 4 2 + 6 2 2 size 12{ { {6 rSup { size 8{2} } +2 rSup { size 8{2} } } over {4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } } } } {} is equivalent to 6 2 + 2 2 ÷ 4 2 + 6 2 2 size 12{ left (6 rSup { size 8{2} } +2 rSup { size 8{2} } right ) div left (4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } right )} {}

Exercises, order of operations

Determine the value of the following:

8 + (32 – 7)

66

(34 + 18 – 2 · 3) + 11

57

8(10) + 4(2 + 3) – (20 + 3 · 15 + 40 – 5)

0

5 · 8 + 42 – 22

52

4(6 2 – 3 3 ) ÷ (4 2 – 4)

9

(8 + 9 · 3) ÷ 7 + 5 · (8 ÷ 4 + 7 + 3 · 5)

125

3 3 + 2 3 6 2 29 + 5 8 2 + 2 4 7 2 3 2 ÷ 8 3 + 1 8 2 3 3 size 12{ { {3 rSup { size 8{3} } +2 rSup { size 8{3} } } over {6 rSup { size 8{2} } - "29"} } +5 left ( { {8 rSup { size 8{2} } +2 rSup { size 8{4} } } over {7 rSup { size 8{2} } - 3 rSup { size 8{2} } } } right ) div { {8 cdot 3+1 rSup { size 8{8} } } over {2 rSup { size 8{3} } - 3} } } {}

7

Module review exercises

For the following problems, find each value.

2 + 3 ( 8 ) size 12{2+3 cdot \( 8 \) } {}

48

1 5 ( 8 8 ) size 12{1 - 5 \( 8 - 8 \) } {}

meaningless

37 1 6 2 size 12{"37" - 1 cdot 6 rSup { size 8{2} } } {}

1

98 ÷ 2 ÷ 7 2 size 12{"98" div 2 div 7 rSup { size 8{2} } } {}

1

( 4 2 2 4 ) 2 3 size 12{ \( 4 rSup { size 8{2} } - 2 cdot 4 \) - 2 rSup { size 8{3} } } {}

0

61 22 + 4 [ 3 ( 10 ) + 11 ] size 12{"61" - "22"+4 \[ 3 cdot \( "10" \) +"11" \] } {}

203

121 4 [ ( 4 ) ( 5 ) 12 ] + 16 2 size 12{"121" - 4 cdot \[ \( 4 \) cdot \( 5 \) - "12" \] + { {"16"} over {2} } } {}

97

2 2 3 + 2 3 ( 6 2 ) ( 3 + 17 ) + 11 ( 6 ) size 12{2 rSup { size 8{2} } cdot 3+2 rSup { size 8{3} } \( 6 - 2 \) - \( 3+"17" \) +"11" \( 6 \) } {}

90

8 ( 6 + 20 ) 8 + 3 ( 6 + 16 ) 22 size 12{ { {8 \( 6+"20" \) } over {8} } + { {3 \( 6+"16" \) } over {"22"} } } {}

29

( 1 + 16 ) 3 7 + 5 ( 12 ) size 12{ { { \( 1+"16" \) - 3} over {7} } +5 \( "12" \) } {}

62

1 6 + 0 8 + 5 2 ( 2 + 8 ) 3 size 12{1 rSup { size 8{6} } +0 rSup { size 8{8} } +5 rSup { size 8{2} } \( 2+8 \) rSup { size 8{3} } } {}

25,001

5 ( 8 2 9 6 ) 2 5 7 + 7 2 4 2 2 4 5 size 12{ { {5 \( 8 rSup { size 8{2} } - 9 cdot 6 \) } over {2 rSup { size 8{5} } - 7} } + { {7 rSup { size 8{2} } - 4 rSup { size 8{2} } } over {2 rSup { size 8{4} } - 5} } } {}

5

6 { 2 8 + 3 } ( 5 ) ( 2 ) + 8 4 + ( 1 + 8 ) ( 1 + 11 ) size 12{6 lbrace 2 cdot 8+3 rbrace - \( 5 \) cdot \( 2 \) + { {8} over {4} } + \( 1+8 \) cdot \( 1+"11" \) } {}

214

26 2 6 + 20 13 size 12{"26"` - `2` cdot ` left lbrace { {6+"20"} over {"13"} } right rbrace } {}

22

( 10 + 5 ) ( 10 + 5 ) 4 ( 60 4 ) size 12{ \( "10"+5 \) ` cdot ` \( "10"+5 \) ` - `4 cdot \( "60" - 4 \) } {}

1

6 2 1 2 3 3 + 4 3 + 2 3 2 5 size 12{ { {6 rSup { size 8{2} } - 1} over {2 rSup { size 8{3} } - 3} } `+` { {4 rSup { size 8{3} } +2` cdot `3} over {2` cdot `5} } } {}

14

51 17 + 7 2 5 12 3 size 12{ { {"51"} over {"17"} } `+`7` - `2` cdot `5` cdot ` left ( { {"12"} over {3} } right )} {}

-30

( 21 3 ) ( 6 1 ) 6 + 4 ( 6 + 3 ) size 12{ \( "21" - 3 \) ` cdot ` \( 6 - 1 \) ` cdot ` left (6 right )+4 \( 6+3 \) } {}

576

( 2 + 1 ) 3 + 2 3 + 1 10 6 2 15 2 [ 2 5 ] 2 5 5 2 size 12{ { { \( 2+1 \) rSup { size 8{3} } +2 rSup { size 8{3} } +1 rSup { size 8{"10"} } } over {6 rSup { size 8{2} } } } ` - ` { {"15" rSup { size 8{2} } - \[ 2` cdot `5 \] rSup { size 8{2} } } over {5` cdot `5 rSup { size 8{2} } } } } {}

0

Questions & Answers

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
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
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Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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