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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

where we get a research paper on Nano chemistry....?
Maira Reply
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
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
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
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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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