# 7.6 Whole numbers: order of operations  (Page 2/2)

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

$\begin{array}{}3+5\cdot 2\\ =8\cdot 2\\ =\text{16}\end{array}$

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

$\begin{array}{}3+5\cdot 2\\ =3+\text{10}\\ =\text{13}\end{array}$

We now have two values for the same expression.

We need a set of rules to guide anyone to one unique value for this kind of expression. Some of these rules are based on convention, while other are forced on up by mathematical logic.

The universally agreed-upon accepted order of operations for evaluating a mathematical expression is as follows:

1. Parentheses (grouping symbols) from the inside out.

By parentheses we mean anything that acts as a grouping symbol, including anything inside symbols such as [  ], {  }, |  |, and $\sqrt{}$ . Any expression in the numerator or denominator of a fraction or in an exponent is also considered grouped, and should be simplified before carrying out further operations.

If there are nested parentheses (parentheses inside parentheses), you work from the innermost parentheses outward.

2. Exponents and other special functions, such as log, sin, cos etc.

3. Multiplications and divisions, from left to right.

4. Additions and subtractions, from left to right.

For example, given: 3 + 15 ÷ 3 + 5 × 2 2+3

The exponent is an implied grouping, so the 2+3 must be evaluated first:

= 3 + 15 ÷ 3 + 5 × 2 5

Now the exponent is carried out:

= 3 +15 ÷ 3 + 5 × 32

Then the multiplication and division, left to right using 15 ÷ 3 = 5 and 5 × 32 = 160:

= 3 + 5 + 160

Finally, the addition, left to right:

= 168

## Examples, order of operation

Determine the value of each of the following.

$\text{21}+3\cdot \text{12}$ .

Multiply first:

= $\text{21}+\text{36}$

= 57

$\left(\text{15}-8\right)+5\left(6+4\right)$ .

Simplify inside parentheses first.

= $7+5\cdot \text{10}$

Multiply.

= $7+\text{50}$

= 57

$\text{63}-\left(4+6\cdot 3\right)+\text{76}-4$ .

Simplify first within the parentheses by multiplying, then adding:

= $\text{63}-\left(4+\text{18}\right)+\text{76}-4$

= $\text{63}-\text{22}+\text{76}-4$

Now perform the additions and subtractions, moving left to right:

= $\text{41}+\text{76}-4$

= $\text{117}-4$

= 113.

$7\cdot 6-{4}^{2}+{1}^{5}$

Evaluate the exponential forms, moving from left to right:

= $7\cdot 6-\text{16}+1$

Multiply 7 · 6:

= $\text{42}-\text{16}+1$

Subtract 16 from 42:

= 26 + 1

= 27.

$6\cdot \left({3}^{2}+{2}^{2}\right)+{4}^{2}$

Evaluate the exponential forms in the parentheses:

= $6\cdot \left(9+4\right)+{4}^{2}$

Add 9 and 4 in the parentheses:

= $6\cdot \left(\text{13}\right)+{4}^{2}$

Evaluate the exponential form ${4}^{2}$ :

= $6\cdot \left(\text{13}\right)+\text{16}$

Multiply 6 and 13:

= $\text{78}+\text{16}$

= 94

$\frac{{6}^{2}+{2}^{2}}{{4}^{2}+6\cdot {2}^{2}}+\frac{{1}^{3}+{8}^{2}}{{\text{10}}^{2}-\text{19}\cdot 5}$ .

= $\frac{\text{36}+4}{\text{16}+6\cdot 4}+\frac{1+\text{64}}{\text{100}-\text{19}\cdot 5}$

= $\frac{\text{36}+4}{\text{16}+\text{24}}+\frac{1+\text{64}}{\text{100}-\text{95}}$

= $\frac{\text{40}}{\text{40}}+\frac{\text{65}}{5}$

= 1+13

= 14

Recall that the bar is a grouping symbol. The fraction $\frac{{6}^{2}+{2}^{2}}{{4}^{2}+6\cdot {2}^{2}}$ is equivalent to $\left({6}^{2}+{2}^{2}\right)÷\left({4}^{2}+6\cdot {2}^{2}\right)$

## Exercises, order of operations

Determine the value of the following:

8 + (32 – 7)

66

(34 + 18 – 2 · 3) + 11

57

8(10) + 4(2 + 3) – (20 + 3 · 15 + 40 – 5)

0

5 · 8 + 42 – 22

52

4(6 2 – 3 3 ) ÷ (4 2 – 4)

9

(8 + 9 · 3) ÷ 7 + 5 · (8 ÷ 4 + 7 + 3 · 5)

125

$\frac{{3}^{3}+{2}^{3}}{{6}^{2}-\text{29}}+5\left(\frac{{8}^{2}+{2}^{4}}{{7}^{2}-{3}^{2}}\right)÷\frac{8\cdot 3+{1}^{8}}{{2}^{3}-3}$

7

## Module review exercises

For the following problems, find each value.

$2+3\cdot \left(8\right)$

48

$1-5\left(8-8\right)$

meaningless

$\text{37}-1\cdot {6}^{2}$

1

$\text{98}÷2÷{7}^{2}$

1

$\left({4}^{2}-2\cdot 4\right)-{2}^{3}$

0

$\text{61}-\text{22}+4\left[3\cdot \left(\text{10}\right)+\text{11}\right]$

203

$\text{121}-4\cdot \left[\left(4\right)\cdot \left(5\right)-\text{12}\right]+\frac{\text{16}}{2}$

97

${2}^{2}\cdot 3+{2}^{3}\left(6-2\right)-\left(3+\text{17}\right)+\text{11}\left(6\right)$

90

$\frac{8\left(6+\text{20}\right)}{8}+\frac{3\left(6+\text{16}\right)}{\text{22}}$

29

$\frac{\left(1+\text{16}\right)-3}{7}+5\left(\text{12}\right)$

62

${1}^{6}+{0}^{8}+{5}^{2}\left(2+8{\right)}^{3}$

25,001

$\frac{5\left({8}^{2}-9\cdot 6\right)}{{2}^{5}-7}+\frac{{7}^{2}-{4}^{2}}{{2}^{4}-5}$

5

$6\left\{2\cdot 8+3\right\}-\left(5\right)\cdot \left(2\right)+\frac{8}{4}+\left(1+8\right)\cdot \left(1+\text{11}\right)$

214

$\text{26}-2\cdot \left\{\frac{6+\text{20}}{\text{13}}\right\}$

22

$\left(\text{10}+5\right)\cdot \left(\text{10}+5\right)-4\cdot \left(\text{60}-4\right)$

1

$\frac{{6}^{2}-1}{{2}^{3}-3}+\frac{{4}^{3}+2\cdot 3}{2\cdot 5}$

14

$\frac{\text{51}}{\text{17}}+7-2\cdot 5\cdot \left(\frac{\text{12}}{3}\right)$

-30

$\left(\text{21}-3\right)\cdot \left(6-1\right)\cdot \left(6\right)+4\left(6+3\right)$

576

$\frac{\left(2+1{\right)}^{3}+{2}^{3}+{1}^{\text{10}}}{{6}^{2}}-\frac{{\text{15}}^{2}-\left[2\cdot 5{\right]}^{2}}{5\cdot {5}^{2}}$

0

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Joseph
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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
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da
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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
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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
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Alexandre
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Rafiq
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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
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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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Mahi
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Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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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
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