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Introduction

In this chapter, you will learn about the short cuts to writing 2 × 2 × 2 × 2 . This is known as writing a number in exponential notation .

Definition

Exponential notation is a short way of writing the same number multiplied by itself many times. For example, instead of 5 × 5 × 5 , we write 5 3 to show that the number 5 is multiplied by itself 3 times and we say “5 to the power of 3”. Likewise 5 2 is 5 × 5 and 3 5 is 3 × 3 × 3 × 3 × 3 . We will now have a closer look at writing numbers using exponential notation.

Exponential Notation

Exponential notation means a number written like

a n

where n is an integer and a can be any real number. a is called the base and n is called the exponent or index .

The n th power of a is defined as:

a n = a × a × × a ( n times )

with a multiplied by itself n times.

We can also define what it means if we have a negative exponent - n . Then,

a - n = 1 a × a × × a ( n times )
Exponentials

If n is an even integer, then a n will always be positive for any non-zero real number a . For example, although - 2 is negative, ( - 2 ) 2 = - 2 × - 2 = 4 is positive and so is ( - 2 ) - 2 = 1 - 2 × - 2 = 1 4 .

Khan academy video on exponents - 1

Khan academy video on exponents-2

Laws of exponents

There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been seen in earlier grades, but we will list all the laws here for easy reference and explain each law in detail, so that you can understand them and not only remember them.

a 0 = 1 a m × a n = a m + n a - n = 1 a n a m ÷ a n = a m - n ( a b ) n = a n b n ( a m ) n = a m n

Exponential law 1: a 0 = 1

Our definition of exponential notation shows that

a 0 = 1 , ( a 0 )

For example, x 0 = 1 and ( 1 000 000 ) 0 = 1 .

Application using exponential law 1: a 0 = 1 , ( a 0 )

  1. 16 0
  2. 16 a 0
  3. ( 16 + a ) 0
  4. ( - 16 ) 0
  5. - 16 0

Exponential law 2: a m × a n = a m + n

Khan academy video on exponents - 3

Our definition of exponential notation shows that

a m × a n = 1 × a × ... × a ( m times ) × 1 × a × ... × a ( n times ) = 1 × a × ... × a ( m + n times ) = a m + n

For example,

2 7 × 2 3 = ( 2 × 2 × 2 × 2 × 2 × 2 × 2 ) × ( 2 × 2 × 2 ) = 2 7 + 3 = 2 10

Interesting fact

This simple law is the reason why exponentials were originally invented. In the days before calculators, all multiplication had to be done by hand with a pencil and a pad of paper. Multiplication takes a very long time to do and is very tedious. Adding numbers however, is very easy and quick to do. If you look at what this law is saying you will realise that it means that adding the exponents of two exponential numbers (of the same base) is the same as multiplying the two numbers together. This meant that for certain numbers, there was no need to actually multiply the numbers together in order to find out what their multiple was. This saved mathematicians a lot of time, which they could use to do something more productive.

Application using exponential law 2: a m × a n = a m + n

  1. x 2 · x 5
  2. 2 3 . 2 4 [Take note that the base (2) stays the same.]
  3. 3 × 3 2 a × 3 2

Exponential law 3: a - n = 1 a n , a 0

Our definition of exponential notation for a negative exponent shows that

a - n = 1 ÷ a ÷ ... ÷ a ( n times ) = 1 1 × a × × a ( n times ) = 1 a n

This means that a minus sign in the exponent is just another way of showing that the whole exponential number is to be divided instead of multiplied.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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