Let $a,b \in \mathbb{C}$ and $n \in \mathbb{N}$. We can present $a$ and $b$ in polar form as
\begin{equation} a = r_a \mathrm{e}^{i \theta_a} \quad \textrm{and} \quad b = r_b \mathrm{e}^{i \theta_b}, \end{equation}
where $r_a,r_b \in \mathbb{R}_+$ are radii, $\theta_a,\theta_b \in [0,2\pi]$ are angles and $i$ is the imaginary unit.
Say we have
\begin{equation} a b^{n+1} = r_a \mathrm{e}^{i \theta_a} r_b^{n+1} \mathrm{e}^{i(n + 1)\theta_b} \end{equation}
and we would like to determine $(a b^{n+1})^{1/n}$. Is it correct to say
\begin{equation} (a b^{n+1})^{1/n} = x_i r_b (r_a r_b)^{1/n} \mathrm{e}^{i \theta_a/n} \mathrm{e}^{i(n + 1)\theta_b/n} = x_i b a^{1/n} b^{1/n} = x_i a^{1/n} b^{1 + 1/n}, \quad i = 1,2,\ldots,n, \end{equation}
where $x_i$ is the $i$th root of unity $x^n = 1$ and $(r_a r_b)^{1/n}$, $a^{1/n}$ and $b^{1/n}$ are principal roots, i.e. the root having smallest positive angle?