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A teacher's guide to fractional exponents.

Start by reminding them of where we are, in the big picture. We started with nothing but the idea that exponents mean “multiply by itself a bunch of times”—in other words, 7 4 means 7 7 7 7 . We went from there to the rules of exponents— x a x b = x a + b and so on—by common sense. Then we said, OK, our definition only works if the exponent is a positive integer. So we found new definitions for zero and negative exponents, but extending down from the positive ones.

Now, we don’t have a definition for fractional exponents. Just as with negative numbers, there are lots of definitions we could make up, but we want to choose one carefully. And we can’t get there using the same trick we used before (you can’t just count and “keep going” and end up at the fractions). But we still have our rules of exponents. So we’re going to see what sort of definition of fractional exponents allows us to keep our rules of exponents.

From there, you just let them start working. I can summarize everything on the assignment in two lines.

  1. The rules of exponents say that ( x 1 2 ) 2 = x . So whatever x 1 2 is, we know that when we square it, we get x . Which means, by definition, that it must be x size 12{ sqrt {x} } {} . Similarly, x 1 3 = x 3 size 12{ nroot { size 8{3} } {x} } {} and so on.
  2. The rules of exponents say that ( x 1 3 ) 2 = x 2 3 . Since we now know that x 1 3 = x 3 size 12{ nroot { size 8{3} } {x} } {} , that means that x 2 3 = x 3 2 size 12{ left ( nroot { size 8{3} } {x} right ) rSup { size 8{2} } } {} . So there you have it.

Thirty seconds, written that way. A whole class period to try to get the students to arrive their on their own, and even there, many of them will require a lot of help to see the point. Toward the end, you may just call the class’s attention to the board and write out the answers. But by the time they leave, you want them to have the following rule: for fractional exponents, the denominator is a root and the numerator is an exponent. And they should have some sense, at least, that this rule followed from the rules of exponents.

There is one more thing I really want them to begin to get. If a problem ends up with 25 size 12{ sqrt {"25"} } {} , you shouldn’t leave it like that. You should call it 5. But if a problem ends up with 2 size 12{ sqrt {2} } {} , you should leave it like that: don’t type it into the calculator and round off. This is also worth explicitly mentioning toward the end.


“Homework: Fractional Exponents”. Mostly this is practicing what they learned in class. The inverse functions are a good exercise: it forces them to review an old topic, but also forces them to practice the current topic. For instance, to find the inverse function of y=x⅔, you write:

x = y 2 3 = y 2 3 size 12{ nroot { size 8{3} } {y rSup { size 8{2} } } } {} . x 3 = y 2 . y = x 3 size 12{ sqrt {x rSup { size 8{3} } } } {} = x 3 2 .

After you go through that exercise a few times, you start to see the pattern that the inverse function actually inverts the exponent. The extra fun comes when you realize that x 0 has no inverse function, just as this rule would predict.

At the end of the homework, they do some graphs—just by plotting points—you will want to make sure they got the shapes right, because this paves the way for the next topic. When going over the homework the next day, sketch the shapes quickly and point out that, on the graph of 2 x , every time you move on to the right, y doubles. On the graph of ( 1 2 ) x , every time you move one to the right, y drops in half.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
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Source:  OpenStax, Advanced algebra ii: teacher's guide. OpenStax CNX. Aug 13, 2009 Download for free at http://cnx.org/content/col10687/1.3
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