<< Chapter < Page Chapter >> Page >
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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set a

Find each product.

x 2 x 5

x 2 + 5 = x 7

Got questions? Get instant answers now!

x 9 x 4

x 9 + 4 = x 13

Got questions? Get instant answers now!

y 6 y 4

y 6 + 4 = y 10

Got questions? Get instant answers now!

c 12 c 8

c 12 + 8 = c 20

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Practice set b

Perform each multiplication in one step.

4 a 3 b 2 9 a 2 b

36 a 5 b 3

Got questions? Get instant answers now!

x 4 4 y 2 2 x 2 7 y 6

56 x 6 y 8

Got questions? Get instant answers now!

( x y ) 3 4 ( x y ) 2

4 ( x y ) 5

Got questions? Get instant answers now!

8 x 4 y 2 x x 3 y 5

8 x 8 y 7

Got questions? Get instant answers now!

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

12 a 10 b 5

Got questions? Get instant answers now!

a n a m a r

a n + m + r

Got questions? Get instant answers now!

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

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
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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.
QuizOver Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask