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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 )

To convince yourself of why this is true, use the fourth exponential law above (division of exponents) and consider what happens when m = n .

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

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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absolutely yes
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it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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