# Thread: complex no. problem no.1

1. ## complex no. problem no.1

question: suppose that
$E = (\frac{1+\sqrt 3 i}2)^n+(\frac{-1+\sqrt 3 i}2)^n$
where n is a natural no.
find the value of E for
a) n = 3k
b) n = 3k +1
c) n = 3k + 2
where k is an odd no.

sorry that i don't type my malfunction workings.
thanks!

2. Hi

You should try using the exponential form of the two complex numbers : it'll make the manipulation of the powers easier.

3. Originally Posted by flyingsquirrel
Hi

You should try using the exponential form of the two complex numbers : it'll make the manipulation of the powers easier.
thanks but i haven't learn exponential yet!

4. OK. You may know De Moivre's formula instead ?

$\left(\cos x+\imath \sin x\right)^n=\cos (nx)+\imath \sin (nx)$

5. Use the following with the above post.
$\frac{1}{2} + i\frac{{\sqrt 3 }}{2} = \cos \left( {\frac{\pi }{3}} \right) + i\sin \left( {\frac{\pi }{3}} \right)\,\& \,\frac{{ - 1}}{2} + i\frac{{\sqrt 3 }}{2} = \cos \left( {\frac{{2\pi }}{3}} \right) + i\sin \left( {\frac{{3\pi }}{3}} \right)
$