Find the fourth roots of 1+sqrt(3)i.
What I did was:
Let z=1+sqrt(3)i, so |z|=r=2 and arg(z)=π/3
So, in polar form: z=2(cos(π/3)+isin(π/3)). Therefore, z^1/4=[2(cos(π/3)+isin(π/3))]^1/4.
Using r^1/n[cos((θ+2πk)/n)+isin((θ+2πk)/n)], where k is an integer, I found the roots.
Is my thinking right?
Thank you in advance for your time and answers.
John
Best Answer
Your reasoning is correct, and your answer is correct so long as you ended up with $4$ distinct complex numbers whose $4$th powers are all $1+i\sqrt3$.