# 3.7 Scientific notation  (Page 2/3)

 Page 2 / 3

$\text{}0.000000000004632\text{}=\text{}4.632×{10}^{-12}$

The decimal point is twelve places to the right of its original position, and the power of 10 is $-12$ .

$0.027=\text{\hspace{0.17em}}2.7×{10}^{-2}$

The decimal point is two places to the right of its original position, and the power of 10 is $-2$ .

## Practice set a

Write the following numbers in scientific notation.

346

$3.46×{10}^{2}$

$72.33$

$7.233×10$

$5387.7965$

$5.3877965×{10}^{3}$

87,000,000

$8.7×{10}^{7}$

179,000,000,000,000,000,000

$1.79×{10}^{20}$

100,000

$1.0×{10}^{5}$

1,000,000

$1.0×{10}^{6}$

$0.0086$

$8.6×{10}^{-3}$

$0.000098001$

$9.8001×{10}^{-5}$

$0.000000000000000054$

$5.4×{10}^{-17}$

$0.0000001$

$1.0×{10}^{-7}$

$0.00000001$

$1.0×{10}^{-8}$

## Scientific form to standard form

A number written in scientific notation can be converted to standard form by reversing the process shown in Sample Set A.

## Converting from scientific notation

To convert a number written in scientific notation to a number in standard form, move the decimal point the number of places prescribed by the exponent on the 10.

## Positive exponent negative exponent

Move the decimal point to the right when you have a positive exponent, and move the decimal point to the left when you have a negative exponent.

## Sample set b

$4.673×{10}^{4}$ .

The exponent of 10 is 4 so we must move the decimal point to the right 4 places (adding $0\text{'}\text{s}$ if necessary). $2.9×{10}^{7}$ .

The exponent of 10 is 7 so we must move the decimal point to the right 7 places (adding $0\text{'}\text{s}$ if necessary).

$2.9×{10}^{7}=\text{29000000}$

$1×{10}^{27}$ .

The exponent of 10 is 27 so we must move the decimal point to the right 27 places (adding $0\text{'}\text{s}$ without a doubt).

$1×{10}^{27}=\text{1,000,000,000,000,000,000,000,000,000}$

$4.21×{10}^{-5}$ .

The exponent of 10 is $-5$ so we must move the decimal point to the left 5 places (adding $0\text{'}\text{s}$ if necessary).

$4.21×{10}^{-5}=\text{0}\text{.0000421}$

$1.006×{10}^{-18}$ .

The exponent of 10 is $-18$ so we must move the decimal point to the left 18 places (adding $0\text{'}\text{s}$ if necessary).

$1.006×{10}^{-18}=0.000000000000000001006$

## Practice set b

Convert the following numbers to standard form.

$9.25×{10}^{2}$

925

$4.01×{10}^{5}$

401000

$1.2×{10}^{-1}$

$0.12$

$8.88×{10}^{-5}$

$0.0000888$

## Multiplying numbers using scientific notation

There are many occasions (particularly in the sciences) when it is necessary to find the product of two numbers written in scientific notation. This is accomplished by using two of the basic rules of algebra.

Suppose we wish to find $\left(a×{10}^{n}\right)\left(b×{10}^{m}\right)$ . Since the only operation is multiplication, we can use the commutative property of multiplication to rearrange the numbers.

$\left(a×{10}^{n}\right)\left(b×{10}^{m}\right)=\left(a×b\right)\left({10}^{n}×{10}^{m}\right)$

Then, by the rules of exponents, ${10}^{n}×{10}^{m}={10}^{n+m}$ . Thus,

$\left(a×{10}^{n}\right)\left(b×{10}^{m}\right)=\left(a×b\right)×{10}^{n+m}$

The product of $\left(a×b\right)$ may not be between 1 and 10, so $\left(a×b\right)×{10}^{n+m}$ may not be in scientific form. The decimal point in $\left(a×b\right)$ may have to be moved. An example of this situation is in Sample Set C, problem 2.

## Sample set c

$\begin{array}{ll}\left(2×{10}^{3}\right)\left(4×{10}^{8}\right)\hfill & =\left(2×4\right)\left({10}^{3}×{10}^{8}\right)\hfill \\ \hfill & =8×{10}^{3+8}\hfill \\ \hfill & =8×{10}^{11}\hfill \end{array}$

$\begin{array}{ll}\left(5×{10}^{17}\right)\left(8.1×{10}^{-22}\right)\hfill & =\left(5×8.1\right)\left({10}^{17}×{10}^{-22}\right)\hfill \\ \hfill & =40.5×{10}^{17-22}\hfill \\ \hfill & =40.5×{10}^{-5}\hfill \end{array}$

We need to move the decimal point one place to the left to put this number in scientific notation.

Thus, we must also change the exponent of 10.

$\begin{array}{l}40.5×{10}^{-5}\hfill \\ 4.05×{10}^{1}×{10}^{-5}\hfill \\ 4.05×\left({10}^{1}×{10}^{-5}\right)\hfill \\ 4.05×\left({10}^{1-5}\right)\hfill \\ 4.05×{10}^{-4}\hfill \end{array}$

Thus,

$\left(5×{10}^{17}\right)\left(8.1×{10}^{-22}\right)=4.05×{10}^{-4}$

how can chip be made from sand
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.

#### Get Jobilize Job Search Mobile App in your pocket Now! By By By Danielle Stephens By OpenStax By Tess Armstrong By Qqq Qqq By Vanessa Soledad By Richley Crapo By OpenStax By Keyaira Braxton By OpenStax By OpenStax