How do you find all solutions to x^4-i=0?

1 Answer
Nov 8, 2016

4 solutions
x_k=cos(pi/8+kpi/2)+isin(pi/8+kpi/2) with k=0,1,2,3

Explanation:

x^4=i
it is better to express the immaginary unit in exponential form
i=e^(ipi/2) in such a way we can rewrite the equation like this
x^4=e^(i(pi/2+2kpi) and as a consequence it must be
x=root4(e^(i(pi/2+2kpi)))=[e^(i(pi/2+2kpi))]^(1/4)=e^(i(pi/8+kpi/2) with k=0,1,2,3