# 3.2 Grouping symbols and the order of operations

 Page 1 / 2
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.

$9+\left(3\cdot 8\right)$

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

$\begin{array}{cc}\hfill 9\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\left(3\phantom{\rule{2px}{0ex}}\cdot \phantom{\rule{2px}{0ex}}8\right)& =\phantom{\rule{2px}{0ex}}9\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\mathrm{24}\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{33}\hfill \end{array}$

Thus,

$9+\left(3\cdot 8\right)=\text{33}$

$\left(\text{10}÷0\right)\cdot 6$

Since $\text{10}÷0$ is undefined, this operation is meaningless, and we attach no value to it. We write, "undefined."

## Practice set a

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

$\text{16}-\left(3\cdot 2\right)$

10

$5+\left(7\cdot 9\right)$

68

$\left(4+8\right)\cdot 2$

24

$\text{28}÷\left(\text{18}-\text{11}\right)$

4

$\left(\text{33}÷3\right)-\text{11}$

0

$4+\left(0÷0\right)$

not possible (indeterminant)

## 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.

$2+\left(8\cdot 3\right)-\left(5+6\right)$

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

$\begin{array}{cc}2+\mathrm{24}-\mathrm{11}\hfill & \text{Now combine left to right.}\\ \mathrm{26}-\mathrm{11}\hfill & \\ \mathrm{15}\hfill & \end{array}$

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

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

$\begin{array}{cc}\mathrm{10}+\left[\mathrm{30}-\mathrm{18}\right]\hfill & \text{Now combine 30 and 18.}\\ \mathrm{10}+\mathrm{12}\hfill & \\ \mathrm{22}\hfill & \end{array}$

## Practice set b

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

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

4

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

93

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

74

${\left\{6-\left[\text{24}÷\left(4\cdot 2\right)\right]\right\}}^{3}$

27

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

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

$\begin{array}{cc}\hfill 3+5\cdot 2& =\phantom{\rule{2px}{0ex}}8\phantom{\rule{2px}{0ex}}\cdot \phantom{\rule{2px}{0ex}}2\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{16}\hfill \end{array}$

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

$\begin{array}{cc}\hfill 3+5\cdot 2& =\phantom{\rule{2px}{0ex}}3\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\mathrm{10}\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{13}\hfill \end{array}$

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.

$\begin{array}{cc}\mathrm{21}+3\cdot \mathrm{12}\hfill & \text{Multiply first.}\hfill \\ \mathrm{21}+\mathrm{36}\hfill & \text{Add.}\hfill \\ \mathrm{57}\hfill & \end{array}$

$\begin{array}{cc}\left(\mathrm{15}-8\right)+5\cdot \left(6+4\right).\hfill & \text{Simplify inside parentheses first.}\hfill \\ 7+5\cdot \mathrm{10}\hfill & \text{Multiply.}\hfill \\ 7+\mathrm{50}\hfill & \text{Add.}\hfill \\ \mathrm{57}\hfill & \end{array}$

$\begin{array}{cc}\mathrm{63}-\left(4+6\cdot 3\right)+\mathrm{76}-4\hfill & \text{Simplify first within the parenthesis by multiplying, then adding.}\hfill \\ \mathrm{63}-\left(4+\mathrm{18}\right)+\mathrm{76}-4\hfill & \hfill \\ \mathrm{63}-\mathrm{22}+\mathrm{76}-4\hfill & \text{Now perform the additions and subtractions, moving left to right.}\hfill \\ \mathrm{41}+\mathrm{76}-4\hfill & \text{Add 41 and 76:}\phantom{\rule{8px}{0ex}}\mathrm{41}+\mathrm{76}=\mathrm{117}.\hfill \\ \mathrm{117}-4\hfill & \text{Subtract 4 from 117:}\phantom{\rule{8px}{0ex}}\mathrm{117}-4=\mathrm{113}.\hfill \\ \mathrm{113}\hfill & \end{array}$

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
7hours 36 min - 4hours 50 min