UNIVERSITY MATH - COMPLEX NO.

On using this equation from De Moivre's Theorem to get the roots of a complex number:

Photo from here: (top photo)

http://math.stackexchange.com/questions/84781/de-moivre-theorem-for-fractional-power-k-n-explanation

I usually get the answer correctly in the first place but how do one gets the principal value?

For example, when the question is

z^4 +8 = 8√3 i

the answers are π/6, 2π/ 3, 14π/ 12 and 20π/12

But 14π/ 12 and 20π/12 is presented finally as -5π/6 and -π/3 finally as their principal values. On the other hand, the answer says that π/6, 2π/ 3 are indeed p.v already so there's no need to change.


I just wanna know how to get 5π/6 and -π/3 from 14π/ 12 and 20π/12.

And the formula of principal value, don't just write

" arg z = argz + 2π n"

I need explanation. Thank you.