I need to find image and kernel of the following homomorphism of groups ($\mathbb{C^*}$ is multiplicative group of complex numbers) $ f: \mathbb{C^*} \rightarrow \mathbb{C^*}$ given by $f(z)=z^n$. I found that $\ker{f} = \{n\text{th roots of unity}\}$. I know that by the homomorphism theorem ${\rm im}\, f\simeq \mathbb{C^*}/\ker{f} $.
I can't find this quotient group. Coset of a given complex number $z \in \mathbb{C^*}$ is (if I'm not wrong) the set of all the nth roots of the complex number $z^n$, but what is the quotient group consisting of all such cosets then (and hence what is the image of the mentioned homomorphism)?