Proving that (z^m)^(1/n)=(z^(1/n))^m

[estro]

Homework Statement

I want to prove something that seems trivial at first:
(z^m)^{1/n}=(z^{1/n})^m, where m and n don't have a common divisor.

The Attempt at a Solution

When I'm using z=re^{i\theta}, I arrive that above is true if and only if the following is true:
e^{ \frac{2\pi \theta mk}{n}}=e^{\frac{2\pi \theta k}{n}}
and this confuses me.

 

[mfb]

When I'm using z=re^{i\theta}, I arrive that above is true if and only if the following is true:
e^{ \frac{2\pi \theta mk}{n}}=e^{\frac{2\pi \theta k}{n}}
and this confuses me.

Then you are doing something wrong.

The equality is true even for complex m,n, so you can consider the more general problem (z^m)^n =? (z^n)^m or (z^m)^n =? z^(mn) and save some fractions.

 

[estro]

This is what puzzlies me:

(z^{\frac {1} {n}})^{m} = r^{\frac {m}{n}}e^{\frac {m\theta + 2m \pi k} {n}i} = <br /> r^{\frac {m}{n}}e^{\frac {m\theta} {n}i}e^{\frac {2m \pi k} {n}i}

(z^m)^{\frac {1} {n}} = r^{\frac {m}{n}}e^{\frac {m\theta + 2 \pi k} {n}i} = <br /> r^{\frac {m}{n}}e^{\frac {m\theta} {n}i}e^{\frac {2 \pi k} {n}i}

How is this possible?

 

[mfb]

You are using n and m with two different meanings, and you don't need that +2pi k there.

 

[estro]

Sorry, but I'm not sure I understand.
m and n are constants while k goes from 0 to n-1.

I do need +2kpi, as I need to prove that both represent the same set of values.

 

[Dick]

Sorry, but I'm not sure I understand.
m and n are constants while k goes from 0 to n-1

I think you are right to to be confused about this. z^(1/n) isn't even a function. It's multivalued. You need to define a branch. If you are naively using the polar form, the answer will depend on the argument you pick for z.

 

[estro]

So how can I prove that both expression represent the following:
https://dl.dropboxusercontent.com/u/27412797/q1.png

 

[Dick]

So how can I prove that both expression represent the following:
https://dl.dropboxusercontent.com/u/27412797/q1.png

You don't do it by assuming that (z^(1/n))^m=(z^m)^(1/n). That's a complex numbers mistake. You can only safely take (z^a)^b=(z^b)^a to be true if a and b are integers. You know from deMoivre that all of those numbers are nth roots of z^m. What I think they want you to show is that that is all of them. For example putting k=n doesn't give you a new root, and that for no two different k in k=0,...n-1 are they the same root.

 

[estro]

The question is formulated like this:
Show that set of values represented by (z^(1/n))^m are the same set of values represented by (z^m)^(1/n).

And both these equivalent to set: https://dl.dropboxusercontent.com/u/27412797/q1.png

 

[Dick]

The question is formulated like this:
Show that set of values represented by (z^(1/n))^m are the same set of values represented by (z^m)^(1/n).

And both these equivalent to set: https://dl.dropboxusercontent.com/u/27412797/q1.png

Alright. The full question has it make a little more sense. I'll give you an example of what can go wrong. Take z=1, m=2 and n=2. Then the values of (1^2)^(1/2) are 1^(1/2) which is +1 or -1. The values of (1^(1/2))^2 are (+1)^2 or (-1)^2 which are both 1. So the value sets aren't equal. Because 2 and 2 aren't relatively prime. That's the problem you are dealing with.

 

[estro]

So I should show that
e^{km\frac {2\pi i} {n}} = e^{k\frac {2\pi i} {n}} because m mod(n) != 0?

 

[Dick]

So I should show that
e^{km\frac {2\pi i} {n}} = e^{k\frac {2\pi i} {n}} because m mod(n) != 0?

Even if that were true, why? I think you should think about it some more. Figure out why it works if n=2 and m=3, those have no common divisor. Choosing z=1 is perfectly general. You've already factored out |z|. You just need to think about the angles.

 

[estro]

e^(2pi*(m/n)k) represent same angle when m and n have common divisor, but when they do the whole expressions becomes equal to 1.But I'm still not sure how to translate this into proving what I need to prove. I'm not even sure why the above is true...

 

[estro]

I think I'm closer to the solution as I understand now that:

e^{2pi \frac {mk}{n} i} = e^{2pi \frac {mk(mod(n))}{n} i}

 

[Dick]

I think I'm closer to the solution as I understand now that:

e^{2pi \frac {mk}{n} i} = e^{2pi \frac {mk(mod(n))}{n} i}

That's a good start. So you know k=0,1,2,...n-1 are the only possible solutions. k=n would just repeat k=0, k=n+1 would just repeat k=1 etc. Now I think you just want to show that they are all different solutions if m and n have no common divisor. This is really turning into a number theory problem.