I am really not doing well on this question
$\displaystyle z^5-i=0$ find all the roots of the polynomial I know there is going to be 5 roots and 2 of them should be -i and +i but i can't figure out the rest?
Do you understand that $\displaystyle r\exp(i\theta)=r(\cos(\theta)+i\sin(\theta))~?$
Now if $\displaystyle \rho = \exp \left( {\frac{{i\pi }}{{10}}} \right)$ then you can see that $\displaystyle \rho^5=i$.
Let $\displaystyle \xi = \exp \left( {\frac{{2\pi i}}{5}} \right)$
Then the five roots you are looking to find are: $\displaystyle \rho\cdot\xi^k~~k=0,1,2,3,4~.$
I got the point of $\displaystyle Z^5-i=0 $$\displaystyle z^5=i$ thus$\displaystyle i$ is one root
but finding the other with the roots I don't understand how $\displaystyle z^5 = i = e^{\pi / 2 + 2 \pi k}$ this is going to help me.
The lightbulb didn't go off yet
I think I am confused because you guys are telling me to use polar form to find my complex roots and oh wait light bulb ... So I am looking for my roots which are x intercepts and then they are the a out of a+bi which means i need to somehow expand my z^5 into a polynomial and wind up with imaginary numbers ?
See I live in montreal and the new gov't is cutting education funding and promoting language issues in an attempted to create a new country .... so we end up with phd math students as teachers and if its a low level course they don't really care because they are more interested in the research they are doing ...
But I love it when you answer my questions, more importantly make me try and figure out the answer myself ...
This was a question from last years final so i figured i should at least try and understand it.