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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: understand the concepts of reciprocals and negative exponents, be able to work with negative exponents.

Overview

  • Reciprocals
  • Negative Exponents
  • Working with Negative Exponents

Reciprocals

Reciprocals

Two real numbers are said to be reciprocals of each other if their product is 1. Every nonzero real number has exactly one reciprocal, as shown in the examples below. Zero has no reciprocal.

4 1 4 = 1. This means that 4 and 1 4 are reciprocals .

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6 1 6 = 1. Hence, 6 and 1 6 are reciprocals .

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2 1 2 = 1. Hence, 2 and 1 2 are reciprocals .

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a 1 a = 1. Hence, a and 1 a are reciprocals if a 0.

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x 1 x = 1. Hence, x and 1 x are reciprocals if x 0.

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x 3 1 x 3 = 1. Hence, x 3 and 1 x 3 are reciprocals if x 0.

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Negative exponents

We can use the idea of reciprocals to find a meaning for negative exponents.

Consider the product of x 3 and x 3 . Assume x 0 .

x 3 x 3 = x 3 + ( 3 ) = x 0 = 1

Thus, since the product of x 3 and x 3 is 1, x 3 and x 3 must be reciprocals.

We also know that x 3 1 x 3 = 1 . (See problem 6 above.) Thus, x 3 and 1 x 3 are also reciprocals.

Then, since x 3 and 1 x 3 are both reciprocals of x 3 and a real number can have only one reciprocal, it must be that x 3 = 1 x 3 .

We have used 3 as the exponent, but the process works as well for all other negative integers. We make the following definition.

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

x n = 1 x n

Sample set a

Write each of the following so that only positive exponents appear.

( 3 a ) 6 = 1 ( 3 a ) 6

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( 5 x 1 ) 24 = 1 ( 5 x 1 ) 24

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( k + 2 z ) ( 8 ) = ( k + 2 z ) 8

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Practice set a

Write each of the following using only positive exponents.

( x y ) 4

1 ( x y ) 4

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( a + 2 b ) 12

1 ( a + 2 b ) 12

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( m n ) ( 4 )

( m n ) 4

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Caution

It is important to note that a n is not necessarily a negative number. For example,

3 2 = 1 3 2 = 1 9 3 2 9

Working with negative exponents

The problems of Sample Set A suggest the following rule for working with exponents:

Moving factors up and down

In a fraction, a factor can be moved from the numerator to the denominator or from the denominator to the numerator by changing the sign of the exponent.

Sample set b

Write each of the following so that only positive exponents appear.

x 2 y 5 . The f a c t o r x 2 can be moved from the numerator to the denominator by changing the exponent 2 to + 2. x 2 y 5 = y 5 x 2

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a 9 b 3 . The f a c t o r b 3 can be moved from the numerator to the denominator by changing the exponent 3 to + 3. a 9 b 3 = a 9 b 3

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a 4 b 2 c 6 . This fraction can be written without any negative exponents by moving the f a c t o r c 6 into the numerator . We must change the 6 to + 6 to make the move legitimate . a 4 b 2 c 6 = a 4 b 2 c 6

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Questions & Answers

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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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
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's the easiest and fastest way to the synthesize AgNP?
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types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
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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
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what is system testing
AMJAD
what is the application of nanotechnology?
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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
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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
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
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silver nanoparticles could handle the job?
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not now but maybe in future only AgNP maybe any other nanomaterials
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I'm interested in Nanotube
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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
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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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