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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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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