# 3.6 Negative exponents

 Page 1 / 2
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

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.

$\begin{array}{ll}4\cdot \frac{1}{4}=1.\hfill & \text{This}\text{\hspace{0.17em}}\text{means}\text{\hspace{0.17em}}\text{that}\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{1}{4}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{reciprocals}.\hfill \end{array}$

$\begin{array}{ll}6\cdot \frac{1}{6}=1.\hfill & \text{Hence,}\text{\hspace{0.17em}}6\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{1}{6}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{reciprocals}.\hfill \end{array}$

$\begin{array}{ll}-2\cdot \frac{-1}{2}=1.\hfill & \text{Hence,}\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}-\frac{1}{2}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{reciprocals}.\hfill \end{array}$

$\begin{array}{ll}a\cdot \frac{1}{a}=1.\hfill & \text{Hence,}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{1}{a}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{reciprocals}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}a\ne 0.\hfill \end{array}$

$\begin{array}{ll}x\cdot \frac{1}{x}=1.\hfill & \text{Hence,}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{1}{x}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{reciprocals}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x\ne 0.\hfill \end{array}$

$\begin{array}{ll}{x}^{3}\cdot \frac{1}{{x}^{3}}=1.\hfill & \text{Hence,}\text{\hspace{0.17em}}{x}^{3}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{1}{{x}^{3}}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{reciprocals}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x\ne 0.\hfill \end{array}$

## 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 $x\ne 0$ .

${x}^{3}\cdot {x}^{-3}={x}^{3+\left(-3\right)}={x}^{0}=1$

Thus, since the product of ${x}^{3}$ and ${x}^{-3}$ is 1, ${x}^{3}$ and ${x}^{-3}$ must be reciprocals.

We also know that ${x}^{3}\cdot \frac{1}{{x}^{3}}=1$ . (See problem 6 above.) Thus, ${x}^{3}$ and $\frac{1}{{x}^{3}}$ are also reciprocals.

Then, since ${x}^{-3}$ and $\frac{1}{{x}^{3}}$ are both reciprocals of ${x}^{3}$ and a real number can have only one reciprocal, it must be that ${x}^{-3}=\frac{1}{{x}^{3}}$ .

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

${x}^{-n}=\frac{1}{{x}^{n}}$

## Sample set a

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

${x}^{-6}=\frac{1}{{x}^{6}}$

${a}^{-1}=\frac{1}{{a}^{1}}=\frac{1}{a}$

${7}^{-2}=\frac{1}{{7}^{2}}=\frac{1}{49}$

${\left(3a\right)}^{-6}=\frac{1}{{\left(3a\right)}^{6}}$

${\left(5x-1\right)}^{-24}=\frac{1}{{\left(5x-1\right)}^{24}}$

${\left(k+2z\right)}^{-\left(-8\right)}={\left(k+2z\right)}^{8}$

## Practice set a

Write each of the following using only positive exponents.

${y}^{-5}$

$\frac{1}{{y}^{5}}$

${m}^{-2}$

$\frac{1}{{m}^{2}}$

${3}^{-2}$

$\frac{1}{9}$

${5}^{-1}$

$\frac{1}{5}$

${2}^{-4}$

$\frac{1}{16}$

${\left(xy\right)}^{-4}$

$\frac{1}{{\left(xy\right)}^{4}}$

${\left(a+2b\right)}^{-12}$

$\frac{1}{{\left(a+2b\right)}^{12}}$

${\left(m-n\right)}^{-\left(-4\right)}$

${\left(m-n\right)}^{4}$

## Caution

It is important to note that ${a}^{-n}$ is not necessarily a negative number. For example,

$\begin{array}{ll}{3}^{-2}=\frac{1}{{3}^{2}}=\frac{1}{9}\hfill & {3}^{-2}\ne -9\hfill \end{array}$

## 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.

$\begin{array}{ll}{x}^{-2}{y}^{5}.\hfill & \text{The}\text{\hspace{0.17em}}factor\text{\hspace{0.17em}}{x}^{-2}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{moved}\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{numerator}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{the}\hfill \\ \hfill & \text{denominator}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{changing}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{exponent}\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}+2.\hfill \\ {x}^{-2}{y}^{5}=\frac{{y}^{5}}{{x}^{2}}\hfill & \hfill \end{array}$

$\begin{array}{ll}{a}^{9}{b}^{-3}.\hfill & \text{The}\text{\hspace{0.17em}}factor\text{\hspace{0.17em}}{b}^{-3}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{moved}\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{numerator}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{the}\hfill \\ \hfill & \text{denominator}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{changing}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{exponent}\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}+3.\hfill \\ {a}^{9}{b}^{-3}=\frac{{a}^{9}}{{b}^{3}}\hfill & \hfill \end{array}$

$\begin{array}{ll}\frac{{a}^{4}{b}^{2}}{{c}^{-6}}.\hfill & \text{This}\text{\hspace{0.17em}}\text{fraction}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{written}\text{\hspace{0.17em}}\text{without}\text{\hspace{0.17em}}\text{any}\text{\hspace{0.17em}}\text{negative}\text{\hspace{0.17em}}\text{exponents}\hfill \\ \hfill & \text{by}\text{\hspace{0.17em}}\text{moving}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}factor\text{\hspace{0.17em}}{c}^{-6}\text{\hspace{0.17em}}\text{into}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{numerator}\text{.}\hfill \\ \hfill & \text{We}\text{\hspace{0.17em}}\text{must}\text{\hspace{0.17em}}\text{change}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}-6\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}+6\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{make}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{move}\text{\hspace{0.17em}}\text{legitimate}\text{.}\hfill \\ \frac{{a}^{4}{b}^{2}}{{c}^{-6}}={a}^{4}{b}^{2}{c}^{6}\hfill & \hfill \end{array}$

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?
no
what is supply
what is opportunity cost
Mizta
The opportunity gained interms of opportunity lost is known as opportunity cost Or The second best alternative use of resources
Mir
forgone alternative: like forgoing Something our of two to buy one
Tam-Waribo
what is macro economic s
macroeconomics is the study of economic as a whole level.
Gafar
meaning of positive science
positive science it is focused on facts and cause and effect and behavioural relationship and include developmental testing in economic theoreis.
Gafar
what is inflation
inflation is the general price increase of goods and services in an economy.
tesfie
Inflation is the persistent rise in the general price level
T-Max
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 .
Ejikeme
inflation is the persistent rise general price level
Habeeb
inflation is the persistent increase in price
Machall
hi
Rafiu
yes
boston
hi
Ayaan
how are you
Ayaan
increase in the general level of price...
what is deflation
Sele
is the gradual decrease of currency exchange in a country.
Gafar
why ecnomics important ? give answer plz
Saifullah
Because is a field of science study that reflects on our day to day activities with human behavior.
ANSU
different between demand and quantity demand
No difference
MansoorAfghan
demand is the overall demand for it
MansoorAfghan
actually theres no difference
MansoorAfghan
quantity demanded is used in Equilibrium of d and s
MansoorAfghan
for evrything else u use deman
MansoorAfghan
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 .☺
Gafar
how to calculate inflation
Explain the factors that have led to high quantity demanded
price of the product increase of price substitute product as people shift to cheap one
Black
what are the methods used by trade union to increase wages of their members?
strike
Pearl
the size of the commodity
Mensah
increase demand of labour decrease supply of labour
Black
keshav
but how do they do it?
Black
by increasing more labour and reduced the suppliers
Mensah
they can not increase labour, they increase demand of labour.
Black
how do they increase demand for labor?
Black
by analyzing the market equilibrium , cost reduction and cost control , savings in time .
yash
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
Black
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.
Black
what is the nature of economics?
economics is a social science since it seeks to solve social problem of scarcity
Jamal
main concerns is the decision individuals make on the allocation of scarce resources among the competing wants
Black
in the short run firm produce a positive as long as the price is larger than what?
what is economic
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
Shaneel
economic s is a social science that deals with human behavior as a relationship between ends and scarce means which has alternative uses
Derokiz
economic is a study of mankind in ordinary business of life
FIDELIS
economics it is the study of social science that deals with human behaviour as relationship between ends and scarce means which have alternative uses
salam
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.
Gafar
what is diminishing returns
what is the difference between calculus linear equation and derivative?
Bti
whats inferior goods?
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 .
yash
What do u understand by the word ENDS in professor Lord L C Robinson definition of Economics?
I understand that ENDS is the unlimited needs of human. But we have limited resources to achieve our unlimited needs/wants. Thank you.
Midhun
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Berger describes sociologists as concerned with
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.