Question about complex numbers

[Chuckster]

Hello guys!

I have a question related to complex numbers.

How would i calculate, for example

$$(\frac{\sqrt{3}+i}{2})^{2010}$$ without using the De Moivre's theorem?

 

[SprucerMoose]

Binomial expansion?

 

[Office_Shredder]

Curl up in a ball and die a little inside.

 

[dalcde]

Use wolfram|alpha to do it for you.

Really, it would be really cumbersome to do it without De Moivre's theorem.

 

[Chuckster]

Binomial expansion?

well, that was my idea originally.
using the facts that
$$i^{1}=i, i^{2}=-1, i^{3}=-i, i^{4}=1$$

i tried to find a way to brake the expression given in the first post into something which could destroy the [te]i[/tex], just like i would do with, ie
$$(1+i)^{2010}=(2i)^{1005}=2^{1005}i$$, but I'm having problems shaking off the real part of the imaginary number.

 

[Office_Shredder]

Using De Moivre's theorem we can immediately observe that if we raise your number to the 12th power you get 1. So prove this with pencil and paper by finding the twelfth power by hand, seeing that it's one, and then you're basically done. You can do this in half the work by just raising it to the sixth power and noticing that you get negative the starting number.

The question then is why are you trying to avoid De Moivre's theorem: because you want avoid using it, or because somebody else is forbidding you from using it in your final solution

 

[olivermsun]

You can do this in half the work by just raising it to the sixth power and noticing that you get negative the starting number.

You can even raise it to the 3rd power using the Binomial theorem and see what you get.