# 1.2 Order of operations for whole numbers

 Page 1 / 2
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:

$\begin{array}{}a×b+c÷d=\left(a×b\right)+\left(c÷d\right)\\ 5×5+\text{20}÷4=\left(5×5\right)+\left(\text{20}÷4\right)\end{array}$

## 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:

= $9+\text{24}$

= $\text{33}$

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.

16 – (3 · 2)

10

5 + (7 · 9)

61

(4 + 8) · 2

24

28 ÷ (18 – 11)

4

(33 ÷ 3) – 11

0

4 + (0 ÷ 0)

meaningless

## 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

$\text{10}+\left[\text{30}-\left(2\cdot 9\right)\right]$

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

= $\text{10}+\left[\text{30}-\text{18}\right]$

Now combine 30 and 18.

= 10 + 12

= 22

## Distributivity

If you see a multiplication outside parentheses like this:

$\begin{array}{}a\left(b+c\right)\\ 3\left(4-3\right)\end{array}$

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

$\begin{array}{}a\left(b+c\right)=\text{ab}+\text{ac}\\ 3\left(4-3\right)=3×4-3×3=\text{12}-9=3\end{array}$

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:

$3\left(4-3\right)=3×\left(1\right)=3$

This can happen with letters too:

$3\left(4a-3a\right)=3×\left(a\right)=3a$

The fact that $a\left(b+c\right)=\text{ab}+\text{ac}$ 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:

$\begin{array}{}\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)\\ =\text{ac}+\text{ad}+\text{bc}+\text{bd}\\ \left(a+3\right)\left(4+d\right)=a\left(4+d\right)+3\left(4+d\right)\\ =4a+\text{ad}+\text{12}+3d\end{array}$

## Multiple grouping symbol exercises

Determine the value of each of the following:

$\left(\text{17}+8\right)+\left(9+\text{20}\right)$

54

$\left(\text{55}-6\right)+\left(\text{13}\cdot 2\right)$

23

$\text{23}+\left(\text{12}÷4\right)+\left(\text{11}\cdot 2\right)$

48

$\text{86}+\left[\text{14}+\left(\text{10}-8\right)\right]$

102

$\text{31}+\left(9+\left[1+\left(\text{35}-2\right)\right]\right)$

74

{6 – [24 ÷ (4 · 2)]} 3

9

## 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 $\text{3}+\text{5}\cdot \text{2}$ . We could do either of two things:

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
Got questions? Join the online conversation and get instant answers!