Find all solutions to $z^5+i=0$ (to $4$ d.p)

Question

The question is find all solutions to $z^5+i=0$ to $4$ d.p. I'm not really sure how to start which is because I'm struggling.

Answer 1

We want to find all the $5$-th roots of $-i$ that in polar form is $e^{\frac{3\pi i }{2}}$

So the roots are $$ e^{\frac{1}{5}\frac{3\pi i }{2}+\frac{2k \pi}{5}}, k=0,...,4 $$

Answer 2

Hint: $i^{1/5}=e^{\frac{\pi i}{10}}$, from Euler's formula.

Now use a primitive $5$th root of unity: $e^{2\pi i/5}$.

So you get $-i,-e^{\frac{9\pi i}{10}},-e^{\frac{13\pi i}{10}},-e^{\frac{17\pi i}{10}},-e^{\frac{21\pi i}{10}}$.

Now you can use a scientific calculator.