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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.Objectives of this module: understand the product and quotient rules for exponents, understand the meaning of zero as an exponent.

Overview

  • The Product Rule for Exponents
  • The Quotient Rule for Exponents
  • Zero as an Exponent

We will begin our study of the rules of exponents by recalling the definition of exponents.

Definition of exponents

If x is any real number and n is a natural number, then

x n = x x x ... x n factors of x

An exponent records the number of identical factors in a multiplication.

Base exponent power

In x n ,

x is the base
n is the exponent
The number represented by x n is called a power .

The term x n is read as " x to the n th."

The product rule for exponents

The first rule we wish to develop is the rule for multiplying two exponential quantities having the same base and natural number exponents. The following examples suggest this rule:

x 2 x 4 = x x x x x x = x x x x x x = x 6 2 + 4 = 6 factors factors

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a a 2 = a a a = a a a = a 3 1 + 2 = 3 factors factors

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Product rule for exponents

If x is a real number and n and m are natural numbers,

x n x m = x n + m

To multiply two exponential quantities having the same base, add the exponents. Keep in mind that the exponential quantities being multiplied must have the same base for this rule to apply.

Sample set a

Find the following products. All exponents are natural numbers.

x 3 x 5 = x 3 + 5 = x 8

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a 6 a 14 = a 6 + 14 = a 20

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y 5 y = y 5 y 1 = y 5 + 1 = y 6

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( x 2 y ) 8 ( x 2 y ) 5 = ( x 2 y ) 8 + 5 = ( x 2 y ) 13

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x 3 y 4 ( x y ) 3 + 4 Since the bases are not the same, the product rule does not apply .

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Practice set a

Find each product.

x 2 x 5

x 2 + 5 = x 7

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x 9 x 4

x 9 + 4 = x 13

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y 6 y 4

y 6 + 4 = y 10

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c 12 c 8

c 12 + 8 = c 20

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( x + 2 ) 3 ( x + 2 ) 5

( x + 2 ) 3 + 5 = ( x + 2 ) 8

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Sample set b

We can use the first rule of exponents (and the others that we will develop) along with the properties of real numbers.

2 x 3 7 x 5 = 2 7 x 3 + 5 = 14 x 8

We used the commutative and associative properties of multiplication. In practice, we use these properties “mentally” (as signified by the drawing of the box). We don’t actually write the second step.

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4 y 3 6 y 2 = 4 6 y 3 + 2 = 24 y 5

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9 a 2 b 6 ( 8 a b 4 2 b 3 ) = 9 8 2 a 2 + 1 b 6 + 4 + 3 = 144 a 3 b 13

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5 ( a + 6 ) 2 3 ( a + 6 ) 8 = 5 3 ( a + 6 ) 2 + 8 = 15 ( a + 6 ) 10

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4 x 3 12 y 2 = 48 x 3 y 2

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The product of four a to the power triangle, and five a to the power star is equal to twenty a to the power 'triangle plus star'.

The bases are the same, so we add the exponents. Although we don’t know exactly what number Sum of a triangle and a star. is, the notation Sum of a triangle and a star. indicates the addition.

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Practice set b

Perform each multiplication in one step.

4 a 3 b 2 9 a 2 b

36 a 5 b 3

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x 4 4 y 2 2 x 2 7 y 6

56 x 6 y 8

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( x y ) 3 4 ( x y ) 2

4 ( x y ) 5

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8 x 4 y 2 x x 3 y 5

8 x 8 y 7

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2 a a a 3 ( a b 2 a 3 ) b 6 a b 2

12 a 10 b 5

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a n a m a r

a n + m + r

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The quotient rule for exponents

The second rule we wish to develop is the rule for dividing two exponential quantities having the same base and natural number exponents.
The following examples suggest a rule for dividing two exponential quantities having the same base and natural number exponents.

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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