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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

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
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
what's the easiest and fastest way to the synthesize AgNP?
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China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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