The $n$ complex $n$th roots of a complex number $z$

If by concise, you are implying brevity which relates the same message we can get rid of your section on re-writing in polar form. When you have $$ \rho^n\exp(in\varphi) = r\exp(i\theta), $$ we can jump straight to $\rho^n = r$ and $n\varphi = \theta + 2k\pi$ for $k,n\in\mathbb{Z}$, so $\rho = r^{1/n}$ and $\varphi = \frac{\theta + 2k\pi}{n}$. Now, we can jump straight to your final thought that the $n$-th roots are $$ w_k = r^{1/n}\exp\biggl[\frac{i(\theta + 2k\pi)}{n}\biggr] $$ where $0\leq k < n-1$. If you want to be clear about the range of $k$, you can say when $k = n$, we have $\theta/n + 2\pi=\theta/n$ since the exponential is periodic $f(\theta) = f(\theta + 2\pi)$.