Hi everyone!
I have this question:
If z = 1 + i√3, use de Moivre's theorem to find z^5 in cartesian form.
I'm a bit lost, do I find all the roots?
The steps I've taken so far are:
1. Convert to polar form
2. Find the roots.
Do I then convert all of the roots back into cartesian?
Cheers.
Thanks for the quick reply!
The answers I have are (they're still in polar form):
z0 = 2^(1/5) . e^(pi/15)i
z1 = 2^(1/5) . e^(7pi/15)i
z2 = 2^(1/5) . e^(13pi/15)i
z3 = 2^(1/5) . e^(19pi/15)i
z4 = 2^(1/5) . e^(5pi/3)i
So I just convert these back to cartesian and I'm done?
I was under the impression that somehow I am supposed to get 1 answer. This method will give me 5 answers. :S
Something like this makes more sense to me:
z = 2.e^(pi/3)i
z^5 = [2.e^(pi/3)i]^5
= 32.e^(5pi/3)i
= 32[cos(5pi/3) + i.sin(5pi/3)]
= 16 - 16i√3
oh, i got confused by the question you asked. you mentioned "roots", in that case you should have 5 answers.
if what the problem actually asked though, was to find z^5, then your last method is the one to choose. do not mix up roots with powers. if the question wanted roots, it would say "roots"