<< Chapter < Page Chapter >> Page >
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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

  

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

Got questions? Get instant answers now!

  

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

Got questions? Get instant answers now!

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

23

Got questions? Get instant answers now!

23 + ( 12 ÷ 4 ) + ( 11 2 ) size 12{"23"+ \( "12" div 4 \) + \( "11" cdot 2 \) } {}

48

Got questions? Get instant answers now!

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

102

Got questions? Get instant answers now!

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

74

Got questions? Get instant answers now!

{6 – [24 ÷ (4 · 2)]} 3

9

Got questions? Get instant answers now!

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 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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Basic math textbook for the community college' conversation and receive update notifications?

Ask