Any Recommendations for Calculating Value Derived from Negative Exponents?

[hemmed]

Hi there,

I feel like I'm posting about exponents in here a lot! Hope this is okay. Have a quick question regarding calculating the value derived from a negative exponent. Here goes. So this is a typical example...

\(\displaystyle \large{\:\frac{1}{\sqrt{4}}\:=\:\frac{1}{\sqrt[2]{2}}\:=\:\frac{1}{2^{\frac{2}{2}}}\:\:=\:\frac{1}{2}}\)

Then, logically you can do the following:


\(\displaystyle {\large{\frac{1}{\frac{1}{2^{-\frac{2}{2}}}}\:=\:\frac{1}{1}\:\cdot \frac{2}{1}^{-\frac{2}{2}}\:=\:2^{-\frac{1}{1}}}}\)

Now again, at this point a lot of my text books would say, hey, flip the statement with the negative exponent around to be the following:

\(\displaystyle {\large\frac{1}{2^{\frac{1}{1}}}}\)

or cleaned up...
\(\displaystyle {\large\frac{1}{2}}\)

But my question is... Are there ever instances where you have to deal with \(\displaystyle {\large{2^{-\frac{1}{1}}}}\) and if so, how do you calculate -1/1? I'm fine with flipping it around so it becomes \(\displaystyle {\large\frac{1}{2^{\frac{1}{1}}}}\) but I want to make sure I can do this in all instances, as at the moment I worry that I can't easily figure out what -1/1 is.

 

[ksdhart2]

Erm... sorry, but I don't think I quite understand what you're trying to ask. -1/1 is simply equal to -1, because you're dividing by 1, which is the identity element in our standard algebra. If your question is about why you can "flip" negative powers and put them in the denominator, that's simply the definition of a negative power. Exponentiation can be thought of as repeated multiplication, so that 2³=2*2*2, 3⁵=3*3*3*3*3, etc. But this only works for integer powers, so we needed to create new definitions to handle negative and rational powers, namely:

\(\displaystyle x^{-n}=\dfrac{1}{x^n}\) and \(\displaystyle x^{\frac{1}{n}}=\sqrt[n]{x}\)

Off the top of my head, I'm not sure if there's another definition to allow for irrational powers or not.

 

[Math for Gwen]

This video I made for my daughter should help you.
https://www.youtube.com/watch?v=fKtMkoFfV-8&list=PLgqHWWmiF4G65zduYRKYa6ixq89Yrj3s4&index=9

Hi there,

I feel like I'm posting about exponents in here a lot! Hope this is okay. Have a quick question regarding calculating the value derived from a negative exponent. Here goes. So this is a typical example...

\(\displaystyle \large{\:\frac{1}{\sqrt{4}}\:=\:\frac{1}{\sqrt[2]{2}}\:=\:\frac{1}{2^{\frac{2}{2}}}\:\:=\:\frac{1}{2}}\)

Then, logically you can do the following:


\(\displaystyle {\large{\frac{1}{\frac{1}{2^{-\frac{2}{2}}}}\:=\:\frac{1}{1}\:\cdot \frac{2}{1}^{-\frac{2}{2}}\:=\:2^{-\frac{1}{1}}}}\)

Now again, at this point a lot of my text books would say, hey, flip the statement with the negative exponent around to be the following:

\(\displaystyle {\large\frac{1}{2^{\frac{1}{1}}}}\)

or cleaned up...
\(\displaystyle {\large\frac{1}{2}}\)

But my question is... Are there ever instances where you have to deal with \(\displaystyle {\large{2^{-\frac{1}{1}}}}\) and if so, how do you calculate -1/1? I'm fine with flipping it around so it becomes \(\displaystyle {\large\frac{1}{2^{\frac{1}{1}}}}\) but I want to make sure I can do this in all instances, as at the moment I worry that I can't easily figure out what -1/1 is.

 

[hemmed]

Erm... sorry, but I don't think I quite understand what you're trying to ask. -1/1 is simply equal to -1, because you're dividing by 1, which is the identity element in our standard algebra. If your question is about why you can "flip" negative powers and put them in the denominator, that's simply the definition of a negative power. Exponentiation can be thought of as repeated multiplication, so that 2³=2*2*2, 3⁵=3*3*3*3*3, etc. But this only works for integer powers, so we needed to create new definitions to handle negative and rational powers, namely:

\(\displaystyle x^{-n}=\dfrac{1}{x^n}\) and \(\displaystyle x^{\frac{1}{n}}=\sqrt[n]{x}\)

Off the top of my head, I'm not sure if there's another definition to allow for irrational powers or not.

Hey,

I think my question was around, how do you handle negative exponents, but this screenshot from Khan Academy really helped me out:



So therefore 2^-1 is really 1*1/2 which is 1/2.