# 5.4 Fractional exponents

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This module provides sample problems designed to develop concepts related to fractional exponents.

On the homework, we demonstrated the rule of negative exponents by building a table. Now, we’re going to demonstrate it another way—by using the rules of exponents.

• ## A

According to the rules of exponents , $\frac{{7}^{3}}{{7}^{5}}$ $7^{\mathrm{\left[ \right]}}$ .
• ## B

But if you write it out and cancel the excess 7s, then $\frac{{7}^{3}}{{7}^{5}}$ = ——.
• ## C

Therefore, since $\frac{{7}^{3}}{{7}^{5}}$ can only be one thing, we conclude that these two things must be equal: write that equation!

Now, we’re going to approach fractional exponents the same way. Based on our rules of exponents , $9^{\left(\frac{1}{2}\right)}^{2}$ =

So, what does that tell us about $9^{\left(\frac{1}{2}\right)}$ ? Well, it is some number that when you square it, you get _______ (*same answer you gave for number 2). So therefore, $9^{\left(\frac{1}{2}\right)}$ itself must be:

Using the same logic, what is $16^{\left(\frac{1}{2}\right)}$ ?

What is $25^{\left(\frac{1}{2}\right)}$ ?

What is $x^{\left(\frac{1}{2}\right)}$ ?

Construct a similar argument to show that $8^{\left(\frac{1}{2}\right)}=2$ .

What is $27^{\left(\frac{1}{3}\right)}$ ?

What is $-1^{\left(\frac{1}{3}\right)}$ ?

What is $x^{\left(\frac{1}{3}\right)}$ ?

What would you expect $x^{\left(\frac{1}{5}\right)}$ to be?

What is $25^{\left(\frac{-1}{2}\right)}$ ? (You have to combine the rules for negative and fractional exponents here!)

Check your answer to #12 on your calculator. Did it come out the way you expected?

OK, we’ve done negative exponents, and fractional exponents—but always with a 1 in the numerator. What if the numerator is not 1?

Using the rules of exponents, $8^{\left(\frac{1}{3}\right)}^{2}=8^{\mathrm{\left[ \right]}}$ .

So that gives us a rule! We know what $8^{\left(\frac{1}{2}\right)}^{2}$ is, so now we know what 8⅔ is.

$8^{\left(\frac{2}{3}\right)}=$

Construct a similar argument to show what $16^{\left(\frac{3}{4}\right)}$ should be.

Check $16^{\left(\frac{3}{4}\right)}$ on your calculator. Did it come out the way you predicted?

Now let’s combine all our rules! For each of the following, say what it means and then say what actual number it is. (For instance, for $9^{\left(\frac{1}{2}\right)}$ you would say it means $\sqrt[]{9}$ so it is 3.)

$8^{\left(\frac{-1}{2}\right)}=$

$8^{\left(\frac{-2}{3}\right)}=$

For these problems, just say what it means. (For instance, $3^{\left(\frac{1}{2}\right)}$ means $\sqrt[]{3}$ , end of story.)

$10^{-4}$

$2^{\left(\frac{-3}{4}\right)}$

$x^{\left(\frac{a}{b}\right)}$

$x^{\left(\frac{\mathrm{-a}}{b}\right)}$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
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
Do you know which machine is used to that process?
s.
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
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
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Advanced algebra ii: activities and homework. OpenStax CNX. Sep 15, 2009 Download for free at http://cnx.org/content/col10686/1.5
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