# 5.1 Real numbers: algebra essentials  (Page 4/35)

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## Differentiating the sets of numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

1. $\sqrt{36}$
2. $\frac{8}{3}$
3. $\sqrt{73}$
4. $-6$
5. $3.2121121112\dots$
N W I Q Q′
a. $\text{\hspace{0.17em}}\sqrt{36}=6$ X X X X
b. $\text{\hspace{0.17em}}\frac{8}{3}=2.\overline{6}$ X
c. $\text{\hspace{0.17em}}\sqrt{73}$ X
d. –6 X X
e. 3.2121121112... X

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

1. $-\frac{35}{7}$
2. $0$
3. $\sqrt{169}$
4. $\sqrt{24}$
5. $4.763763763\dots$
N W I Q Q'
a. $\text{\hspace{0.17em}}-\frac{35}{7}$ X X
b. 0 X X X
c. $\text{\hspace{0.17em}}\sqrt{169}$ X X X X
d. $\text{\hspace{0.17em}}\sqrt{24}$ X
e. 4.763763763... X

## Performing calculations using the order of operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, $\text{\hspace{0.17em}}{4}^{2}=4\cdot 4=16.\text{\hspace{0.17em}}$ We can raise any number to any power. In general, the exponential notation     $\text{\hspace{0.17em}}{a}^{n}\text{\hspace{0.17em}}$ means that the number or variable $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is used as a factor $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ times.

In this notation, $\text{\hspace{0.17em}}{a}^{n}\text{\hspace{0.17em}}$ is read as the n th power of $\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is called the base    and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is called the exponent     . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, $\text{\hspace{0.17em}}24+6\cdot \frac{2}{3}-{4}^{2}\text{\hspace{0.17em}}$ is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations    . This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

$24+6\cdot \frac{2}{3}-{4}^{2}$

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify $\text{\hspace{0.17em}}{4}^{2}\text{\hspace{0.17em}}$ as 16.

$\begin{array}{l}\hfill \\ \begin{array}{l}24+6\cdot \frac{2}{3}-{4}^{2}\hfill \\ 24+6\cdot \frac{2}{3}-16\hfill \end{array}\hfill \end{array}$

Next, perform multiplication or division, left to right.

$\begin{array}{l}\hfill \\ \begin{array}{l}24+6\cdot \frac{2}{3}-16\hfill \\ 24+4-16\hfill \end{array}\hfill \end{array}$

Lastly, perform addition or subtraction, left to right.

Therefore, $\text{\hspace{0.17em}}24+6\cdot \frac{2}{3}-{4}^{2}=12.$

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

## Order of operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses)
E (xponents)
M (ultiplication) and D (ivision)
A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

1. Simplify any expressions within grouping symbols.
2. Simplify any expressions containing exponents or radicals.
3. Perform any multiplication and division in order, from left to right.
4. Perform any addition and subtraction in order, from left to right.

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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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