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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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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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