Thread: Complex Numbers in Exponential form

Hey guys, got some questions which I need help in:

1) Express w = (10-2.(sq. root 3)i)/(1-3(sq.root 3)i) in the exponential form and hence or otherwise find the smallest +ve integer n such that w^n is a real number.

I expressed in exponential form and got w = 2e^i(\pi /3) but I got no idea how to do the 2nd part...

Another similar question which I need help in is this:

2)Given that z = (sq. root 3) + i, express z in the exponential form. Hence, find the real value of k such that z^5 + kz* = 0

Again, I only know how to express in exponential form... z = 2e^i(\pi /6)

Please help! Thanks in advance!

You should know that . What power will you have to take to in order to get your exponent equal to ?

Originally Posted by Blizzardy

1) Express w = (10-2.(sq. root 3)i)/(1-3(sq.root 3)i) in the exponential form and hence or otherwise find the smallest +ve integer n such that w^n is a real number.
I expressed in exponential form and got w = 2e^i(\pi /3) but I got no idea how to do the 2nd part...

Clearly the answer to part 2 is 3.
Do you see why

Many thanks guys! Yupyup I understand Q1. =) But I still can't do Q2.

2)Given that z = (sq. root 3) + i, express z in the exponential form. Hence, find the real value of k such that z^5 + kz* = 0

So, z = 2e^i(\pi /6)
After substitution, I get: 2^5e^i(5pi/6) + ke^i(-pi/6) = 0
But how do I continue?

Originally Posted by Blizzardy

Many thanks guys! So, z = 2e^i(\pi /6)
After substitution, I get: 2^5e^i(5pi/6) + ke^i(-pi/6) = 0
But how do I continue?

Just notice that .