What is $\sqrt{(-1)^2}$

Question

This question is primarily terminology based. In that $\sqrt{}$ denotes the principal square root. Here are two reasoning

1. $\sqrt{(-1)^2}=1$ since $\sqrt{(-1)^2}=\sqrt{1}$ which we know has a principal square root of $1$.

2. Or $\sqrt{(-1)^2}=-1$ since $\sqrt{(e^{\pi i})^2}=\sqrt{e^{2\pi i}}=e^{\pi i}=-1$

Which reasoning is correct and why? Also if possible can you leave a source.

Answer 1

$\sqrt{(-1)^2} = \sqrt{1} = 1$

Answer 2

The first one is correct.

In the second one, $\sqrt{e^{2\pi i}}=e^{\pi i}$ is wrong. Should be $\sqrt{e^{2\pi i}}=|e^{\pi i}|=1$ (This stands of course under that assumption of knowing $e^{\pi i}$ is real).

Answer 3

$(e^{2\pi i})=(\cos(2\pi)+i\sin(2\pi))=1+0i=1$

$\sqrt{1}=1$

Answer 4

according to the absolute definition
$$\sqrt{x^2}=|x|=|-1|=1$$