Hi Can I just get someone to confirm my answer is right for the following:
I want to find all values of $1^{\frac{1}{5}}$. So I worked these out to be $1e^{0i}, 1e^{\frac{2\pi i}{5}}, 1e^{\frac{2\pi i}{10}}, 1e^{\frac{2\pi i}{15}}, 1e^{\frac{2\pi i}{20}}$. Just making sure my angles are correct? I've just added $\frac{2\pi i}{5}$ to each preceeding angle.
Secondly, I want to find the linear factors of $P(z) = 1+z+z^2+...+z^8$. So, just to be sure, I am looking for the $9th$ roots of unity which are $(z-e^{{\frac{2\pi ki}{9}}})$ where k = 0,...,8?
Thank you
For the first one, you have the right idea but have not made it work properly. The right answer is $1=e^{\frac{0\pi i}{5}}$, $e^{\frac{2\pi i}{5}}$, $e^{\frac{4\pi i}{5}}$, $e^{\frac{6\pi i}{5}}$, $e^{\frac{8\pi i}{5}}$.
For the second one: your answer would be correct for the polynomial $z^9-1=(z-1)\cdot P(z)$. For $P(z)$ you must omit the factor $z-1$, i.e. $k$ will go from $1$ to $8$, rather than from $0$ to $8$.