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In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first.

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.

For example:

(5 · 5) + 20 = 45

whereas:

5 · (5 + 20) = 125

If there are no parentheses, you should always do multiplications and divisions first followed by additions and subtractions. You can always put your own parentheses into equations using this rule to make things easier for yourself, for example:

a × b + c ÷ d = ( a × b ) + ( c ÷ d ) 5 × 5 + 20 ÷ 4 = ( 5 × 5 ) + ( 20 ÷ 4 ) alignl { stack { size 12{a times b+c div d= \( a times b \) + \( c div d \) } {} #size 12{5 times 5+"20" div 4= \( 5 times 5 \) + \( "20" div 4 \) } {} } } {}

Grouping symbols examples

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

Example 1

9 + (3 · 8)

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

= 9 + 24 size 12{9+"24"} {}

Then add the terms:

= 33 size 12{"33"} {}

Thus, 9 + (3 · 8) = 33.

Example 2

(10 ÷ 0) · 6

Since (10 ÷ 0) is undefined, this operation is meaningless, and we attach no value to it. We write, “meaningless.”

Grouping symbols exercises

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

4 + (0 ÷ 0)

meaningless

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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.

Multiple grouping symbol examples

Determine the value of each of the following.

2 + (8 · 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 ] size 12{"10"+ \[ "30" - "18" \] } {}

Now combine 30 and 18.

= 10 + 12

= 22

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Distributivity

If you see a multiplication outside parentheses like this:

a ( b + c ) 3 ( 4 3 ) alignl { stack { size 12{a \( b+c \) } {} #size 12{3 \( 4 - 3 \) } {} } } {}

then it means you have to multiply each part inside the parentheses by the number outside:

a ( b + c ) = ab + ac 3 ( 4 3 ) = 3 × 4 3 × 3 = 12 9 = 3 alignl { stack { size 12{a \( b+c \) = ital "ab"+ ital "ac"} {} #size 12{3 \( 4 - 3 \) =3 times 4 - 3 times 3="12" - 9=3} {} } } {}

Sometimes you can simplify everything inside the parentheses into a single term. In fact, in the above example, it would have been smarter to have done this:

3 ( 4 3 ) = 3 × ( 1 ) = 3 size 12{3 \( 4 - 3 \) =3 times \( 1 \) =3} {}

This can happen with letters too:

3 ( 4a 3a ) = 3 × ( a ) = 3a size 12{3 \( 4a - 3a \) =3 times \( a \) =3a} {}

The fact that a ( b + c ) = ab + ac size 12{a \( b+c \) = ital "ab"+ ital "ac"} {} is know as the distributive property.

If there are two sets of parentheses multiplied by each other, then you can do it one step at a time:

( a + b ) ( c + d ) = a ( c + d ) + b ( c + d ) = ac + ad + bc + bd ( a + 3 ) ( 4 + d ) = a ( 4 + d ) + 3 ( 4 + d ) = 4a + ad + 12 + 3d alignl { stack { size 12{ \( a+b \) \( c+d \) =a \( c+d \) +b \( c+d \) } {} #size 12{ {}= ital "ac"+ ital "ad"+ ital "bc"+ ital "bd"} {} # size 12{ \( a+3 \) \( 4+d \) =a \( 4+d \) +3 \( 4+d \) } {} #size 12{ {}=4a+ ital "ad"+"12"+3d} {} } } {}

Multiple grouping symbol exercises

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"+ \( "12" div 4 \) + \( "11" cdot 2 \) } {}

48

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

102

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

74

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{6 – [24 ÷ (4 · 2)]} 3

9

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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:

Questions & Answers

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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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
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Source:  OpenStax, Basic math textbook for the community college. OpenStax CNX. Jul 04, 2009 Download for free at http://cnx.org/content/col10726/1.1
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