# 5.2 Exponents and scientific notation  (Page 6/9)

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Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

$4.7\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-13}$

## Scientific notation

A number is written in scientific notation    if it is written in the form $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{n},$ where $\text{\hspace{0.17em}}1\le |a|<10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is an integer.

## Converting standard notation to scientific notation

Write each number in scientific notation.

1. Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
2. Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m
3. Number of stars in Andromeda Galaxy: 1,000,000,000,000
4. Diameter of electron: 0.00000000000094 m
5. Probability of being struck by lightning in any single year: 0.00000143

Write each number in scientific notation.

1. U.S. national debt per taxpayer (April 2014): $152,000 2. World population (April 2014): 7,158,000,000 3. World gross national income (April 2014):$85,500,000,000,000
4. Time for light to travel 1 m: 0.00000000334 s
5. Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715
1. $1.52×{10}^{5}$
2. $7.158×{10}^{9}$
3. $8.55×{10}^{13}$
4. $3.34×{10}^{-9}$
5. $7.15×{10}^{-8}$

## Converting from scientific to standard notation

To convert a number in scientific notation    to standard notation, simply reverse the process. Move the decimal $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ places to the right if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is positive or $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ places to the left if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is negative and add zeros as needed. Remember, if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is positive, the value of the number is greater than 1, and if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is negative, the value of the number is less than one.

## Converting scientific notation to standard notation

Convert each number in scientific notation to standard notation.

1. $3.547\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{14}$
2. $-2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{6}$
3. $7.91\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-7}$
4. $-8.05\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-12}$

Convert each number in scientific notation to standard notation.

1. $7.03\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{5}$
2. $-8.16\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{11}$
3. $-3.9\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-13}$
4. $8\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-6}$
1. $703,000$
2. $-816,000,000,000$
3. $-0.000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}39$
4. $0.000008$

## Using scientific notation in applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around $\text{\hspace{0.17em}}1.32\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{21}\text{\hspace{0.17em}}$ molecules of water and 1 L of water holds about $\text{\hspace{0.17em}}1.22\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{4}\text{\hspace{0.17em}}$ average drops. Therefore, there are approximately $\text{\hspace{0.17em}}3\cdot \left(1.32\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{21}\right)\cdot \left(1.22\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{4}\right)\approx 4.83\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{25}\text{\hspace{0.17em}}$ atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
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
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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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