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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

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
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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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
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Google
da
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Bhagvanji
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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
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yes
narayan
what is variations in raman spectra for nanomaterials
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ya I also want to know the raman spectra
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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
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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
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Rafiq
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Anam
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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Bob Reply
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Bob
The nanotechnology is as new science, to scale nanometric
brayan
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Damian
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Damian Reply
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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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