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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

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
learn
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
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
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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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