maybe this question is very simple and clear and trivial to everbody.
but right now i'm not sure.
the equotation
$$ x^\frac{1}{2} = \sqrt{x} $$ is only true whenever $ x \geq 0 $ right?
the square root is only defined for positive numbers in "real numbers".
so whats about
$$ x^{2^\frac{1}{2}} = x = \sqrt{x^2} $$
if x would be negative, this equotation wouldn't be true. So my question is, when is it allowed to multiply powers, and when not?
The question is, is the square root always $x^{\frac 12}$ and only defined in positve numbers, or is it only allowed to $x^{\frac 12} = \sqrt{x}$ whenever we are in positive numbers?