# 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.

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

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

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

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
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.