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

Nov 8, 2016

4 solutions
${x}_{k} = \cos \left(\frac{\pi}{8} + k \frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{8} + k \frac{\pi}{2}\right)$ with $k = 0 , 1 , 2 , 3$

#### Explanation:

${x}^{4} = i$
it is better to express the immaginary unit in exponential form
$i = {e}^{i \frac{\pi}{2}}$ 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 $k = 0 , 1 , 2 , 3$