The root test for convergence of a complex power series is given as
$$\lim_{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} = L$$
If $a_n = \frac1{(1+i)^n}$ then I read that when applying the root test I can just remove the powers since they cancel out:
$$\lim_{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} = \lim_{n \rightarrow \infty}\sqrt[n]{\left|\frac1{(1+i)^n}\right|} = \lim_{n \rightarrow \infty} \left|\frac1{(1+i)}\right|$$
Why is it ok to cancel the root outside the absolute function with the power inside the absolute function? For any given $n$ the expression might be negative so I feel this shouldn't be possible.
Thank you.
This is because $$|z^n|=|z|^n\qquad\forall z\in\mathbb{C},n\in\mathbb{N}$$ which can be proven by considering both sides in exponential form.