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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.


  • Reciprocals
  • Negative Exponents
  • Working with Negative Exponents



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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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

Would you expect the kinked demand curve to be more extreme (like a right angle) or less extreme (like a normal demand curve) if each firm in the cartel produces a near-identical product like OPEC and petroleum? What if each firm produces a somewhat different product?
James Reply
what is supply
Mizta Reply
what is opportunity cost
The opportunity gained interms of opportunity lost is known as opportunity cost Or The second best alternative use of resources
forgone alternative: like forgoing Something our of two to buy one
what is macro economic s
Addo Reply
macroeconomics is the study of economic as a whole level.
meaning of positive science
Sumit Reply
positive science it is focused on facts and cause and effect and behavioural relationship and include developmental testing in economic theoreis.
what is inflation
Sama Reply
inflation is the general price increase of goods and services in an economy.
Inflation is the persistent rise in the general price level
inflation is characterized by increase in the general price of goods and services. when there is too much money in circulation. increase in demand of goods pursuing fewer goods. when purchasing power of money decreases .
inflation is the persistent rise general price level
inflation is the persistent increase in price
how are you
increase in the general level of price...
what is deflation
is the gradual decrease of currency exchange in a country.
why ecnomics important ? give answer plz
Because is a field of science study that reflects on our day to day activities with human behavior.
different between demand and quantity demand
Farhan Reply
No difference
demand is the overall demand for it
actually theres no difference
quantity demanded is used in Equilibrium of d and s
for evrything else u use deman
the difference of it is that when demand simply denotes the willingness and a person's ability to purchase. And as against quantity demand represent the amount of an economic good or services desire by a consumer at a fixed price .☺
how to calculate inflation
Richard Reply
Explain the factors that have led to high quantity demanded
Ogwang Reply
price of the product increase of price substitute product as people shift to cheap one
what are the methods used by trade union to increase wages of their members?
Black Reply
the size of the commodity
increase demand of labour decrease supply of labour
I do support your answer Jackel.
but how do they do it?
by increasing more labour and reduced the suppliers
they can not increase labour, they increase demand of labour.
how do they increase demand for labor?
by analyzing the market equilibrium , cost reduction and cost control , savings in time .
decreasing supply of labour are achieved through training and certification that require for you to employed, you must have certificate, also trade union encouraged government to restrict migration into the country causing shortage of labour supply. Note that the aim of union is to enhance life
objective of union: better working conditions, liveable wage, protect member from unfair treatment which are done through negotiations betweens representative and management. known as collective bargaining.
what is the nature of economics?
Tyscar Reply
economics is a social science since it seeks to solve social problem of scarcity
main concerns is the decision individuals make on the allocation of scarce resources among the competing wants
in the short run firm produce a positive as long as the price is larger than what?
yoel Reply
what is economic
Bah Reply
economics is the study of managing the resources in order to maximize the needs and satisfy the wants to a great extent in a regulated set-up..
One explanation for deviation when there is no impact on balance of trade
economic s is a social science that deals with human behavior as a relationship between ends and scarce means which has alternative uses
economic is a study of mankind in ordinary business of life
economics it is the study of social science that deals with human behaviour as relationship between ends and scarce means which have alternative uses
Economic is the use of scarce recourses to attain economic dough effectively and efficiently.
economics is the study of how humans make decisions in the face of scarcity. Eg. family decision, individual decision, and societal decision.
what is diminishing returns
Blessed Reply
what is the difference between calculus linear equation and derivative?
whats inferior goods?
jaamac Reply
Good having low quality , also known as giffin goods. When income increases people shift to better quality goods . Hence having a negative effect on inferior goods rather than positive relation ( ie when income increases demand increases but not in case of inferior goods ) example wheat and bajra .
What do u understand by the word ENDS in professor Lord L C Robinson definition of Economics?
Kaba Reply
I understand that ENDS is the unlimited needs of human. But we have limited resources to achieve our unlimited needs/wants. Thank you.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
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Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
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The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
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Berger describes sociologists as concerned with
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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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