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3.2 Grouping symbols and the order of operations

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses grouping symbols and the order of operations. By the end of the module students should be able to understand the use of grouping symbols, understand and be able to use the order of operations and use the calculator to determine the value of a numerical expression.

Section overview

  • Grouping Symbols
  • Multiple Grouping Symbols
  • The Order of Operations
  • Calculators

Grouping symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:

( ), [ ], { },

Parentheses : ( )
Brackets : [ ]
Braces : { }
Bar :

In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first. If possible, we perform operations inside grouping symbols first.

Sample set a

If possible, determine the value of each of the following.

9 + ( 3 ⋅ 8 ) size 12{9+ \( 3 cdot 8 \) } {}

Since 3 and 8 are within parentheses, they are to be combined first.

9 + ( 3 ⋅ 8 ) = 9 + 24 = 33

Thus,

9 + ( 3 ⋅ 8 ) = 33 size 12{9+ \( 3 cdot 8 \) ="33"} {}

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( 10 ÷ 0 ) ⋅ 6 size 12{ \( "10"÷0 \) cdot 6} {}

Since 10 ÷ 0 size 12{"10" div 0} {} is undefined, this operation is meaningless, and we attach no value to it. We write, "undefined."

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Practice set a

If possible, determine the value of each of the following.

16 − ( 3 ⋅ 2 ) size 12{"16"- \( 3 cdot 2 \) } {}

10

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5 + ( 7 ⋅ 9 ) size 12{5+ \( 7 cdot 9 \) } {}

68

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( 4 + 8 ) ⋅ 2 size 12{ \( 4+8 \) cdot 2} {}

24

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28 ÷ ( 18 − 11 ) size 12{"28"÷ \( "18"-"11" \) } {}

4

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( 33 ÷ 3 ) − 11 size 12{ \( "33"÷3 \) -"11"} {}

0

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4 + ( 0 ÷ 0 ) size 12{4+ \( 0÷0 \) } {}

not possible (indeterminant)

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Multiple grouping symbols

When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations within the innermost set first.

Sample set b

Determine the value of each of the following.

2 + ( 8 ⋅ 3 ) − ( 5 + 6 ) size 12{2+ \( 8 cdot 3 \) - \( 5+6 \) } {}

Combine 8 and 3 first, then combine 5 and 6.

2 + 24 - 11 Now combine left to right. 26 - 11 15

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10 + [ 30 − ( 2 ⋅ 9 ) ] size 12{"10"+ \[ "30"- \( 2 cdot 9 \) \] } {}

Combine 2 and 9 since they occur in the innermost set of parentheses.

10 + [ 30 - 18 ] Now combine 30 and 18. 10 + 12 22

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Practice set b

Determine the value of each of the following.

( 17 + 8 ) + ( 9 + 20 ) size 12{ \( "17"+8 \) + \( 9+"20" \) } {}

54

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( 55 − 6 ) − ( 13 ⋅ 2 ) size 12{ \( "55"-6 \) - \( "13" cdot 2 \) } {}

23

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23 + ( 12 ÷ 4 ) − ( 11 ⋅ 2 ) size 12{"23"+ \( "12"÷4 \) - \( "11" cdot 2 \) } {}

4

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86 + [ 14 ÷ ( 10 − 8 ) ] size 12{"86"+ \[ "14"÷ \( "10"-8 \) \] } {}

93

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31 + { 9 + [ 1 + ( 35 − 2 ) ] } size 12{"31"+ lbrace 9+ \[ 1+ \( "35"-2 \) \] rbrace } {}

74

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{ 6 − [ 24 ÷ ( 4 ⋅ 2 ) ] } 3 size 12{ lbrace 6- \[ "24"÷ \( 4 cdot 2 \) \] rbrace rSup { size 8{3} } } {}

27

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The order of operations

Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of 3 + 5 ⋅ 2 size 12{3+5 cdot 2} {} . We could do either of two things:

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

3 + 5 ⋅ 2 = 8 ⋅ 2 = 16

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

3 + 5 ⋅ 2 = 3 + 10 = 13

We now have two values for one number. To determine the correct value, we must use the accepted order of operations .

    Order of operations

  1. Perform all operations inside grouping symbols, beginning with the innermost set, in the order 2, 3, 4 described below,
  2. Perform all exponential and root operations.
  3. Perform all multiplications and divisions, moving left to right.
  4. Perform all additions and subtractions, moving left to right.

Sample set c

Determine the value of each of the following.

21 + 3 ⋅ 12 Multiply first. 21 + 36 Add. 57

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( 15 - 8 ) + 5 ⋅ ( 6 + 4 ) . Simplify inside parentheses first. 7 + 5 ⋅ 10 Multiply. 7 + 50 Add. 57

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63 - ( 4 + 6 ⋅ 3 ) + 76 - 4 Simplify first within the parenthesis by multiplying, then adding. 63 - ( 4 + 18 ) + 76 - 4 63 - 22 + 76 - 4 Now perform the additions and subtractions, moving left to right. 41 + 76 - 4 Add 41 and 76: 41 + 76 = 117 . 117 - 4 Subtract 4 from 117: 117 - 4 = 113 . 113

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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