I have just started learning complex numbers and am now confused about one property.
We know that $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$ is only valid if both $a$ and $b$ are not negative simultaneously. My teacher told me that the relationship breaks if both the numbers are negative to solve the following contradiction.
$\sqrt{-1}\cdot \sqrt{-1}=\sqrt{(-1)(-1)}=1$
Now, that makes me wonder if the relationship also holds if $a$ or $b$ are complex numbers with non-zero imaginary parts. Any help would be greatly appreciated.
Taking "to the $0.5$" power makes much less sense for complex numbers than for real numbers. For real numbers, you can get away with defining square roots by choosing the positive square root. But in the complex numbers, how do you decide if $-3+4i$ or $3-4i$ is the square root of $-7 - 24i$? You can't do this in a continuous way, and so asking about such relations is difficult to make sense of, since you need to specify how you're square rooting.