Math Help - Roots of Unity

1. Roots of Unity

Hi,
Roots of unity is a very easy topic for me, though i am misunderstanding one small part of it which i wish to rectify.
Here's an example:
$z^{\frac{6}{7}}=e^{\frac{i\pi}{5}}$
$z^{6}=e^{\frac{7i\pi}{5}+2\pi in}$
$z=e^{\frac{7i\pi}{30}+\frac{2\pi in}{6}}$

Where n is an integer, i is the complex number and e an exponential.

Now i undertand the use of 2pi i n, as its roots of unity and therefore will go back to being equal to 1.

However i don't understand why we pluck in the 2 pi i n in the second step, rather than the first or the last. By raising to the power that induces the 2 pi i n to appear? Why?

Hope you can help me

2. Originally Posted by imagemania
Hi,
Roots of unity is a very easy topic for me, though i am misunderstanding one small part of it which i wish to rectify.
Here's an example:
$z^{\frac{6}{7}}=e^{\frac{i\pi}{5}}$
$z^{6}=e^{\frac{7i\pi}{5}+2\pi in}$
$z^{6}=e^{\frac{7i\pi}{30}+\frac{2\pi in}{6}}$
Where n is an integer, i is the complex number and e an exponential.
Now i undertand the use of 2pi i n, as its roots of unity and therefore will go back to being equal to 1.
However i don't understand why we pluck in the 2 pi i n in the second step, rather than the first or the last. By raising to the power that induces the 2 pi i n to appear? Why?
Can you please explain how this question has anything to do with roots of unity/
It does not make any sense to me.

3. [Ignore- it double posted for some reason] :\

4. Well it might not be, but its under the same area in my notes, the section starts of with:
$z=e^{2\pi ik} = 1$ where k is an integer.
$z^{\frac{1}{n}}=e^{\frac{2\pi ik}{n} = 1^{\frac{1}{n}}$
Then it extrapolates by bringing in n roots

Oh i also made a mistake there, the last z should not be raised to the 6.

5. Originally Posted by imagemania
Well it might not be, but its under the same area in my notes, the section starts of with:
$z=e^{2\pi ik} = 1$ where k is an integer.
$z^{\frac{1}{n}}=e^{\frac{2\pi ik}{n} = 1^{\frac{1}{n}}$
Then it extrapolates by bringing in n rootsOh i also made a mistake there, the last z should not be raised to the 6.
That is still nonsense as far as I can read it.
Why don't you simply post a clear question?