Is there a sense in which $4^{2} = -16$? Here's my train of thought:
I. $4^2=4^{(4/2)}$
II. $4^{(4/2)}=(4^4)^{(1/2)}$
III. $(4^4)^{(1/2)}=256^{(1/2)}$
IV. $256^{(1/2)}=\sqrt{256}$
square root of 256 has 2 solutions: 16 and -16
Can you explain me where I made the mistake and why am I not allowed to do whatever I did?
I thought that maybe (II) is wrong and only $4^{(4/2)}=(4^{(1/2)})^4$ is true,
but if $(x^a)^b=x^{(a\cdot b)}$ and numbers can switch places during multiplication and this does not change the outcome then:
$x^{(a\cdot b)}=x^{(b\cdot a)}$, and $(x^b)^a=x^{(b\cdot a)}$
Also, please don't insult me in the replies. I feel stupid enough already for not getting this.

Your I to IV are all correct. Here is what is incorrect: "square root of $256$ has two solutions." That is not true.
It is true that the equation $x^2 = 256$ has two solutions. But by convention, the square root  (written as $\sqrt{256}$ or $\sqrt{256}$ or $256^{1/2}$) is defined to be the non negative solution. It would have been possible to define it to be the negative one, but nobody does that. But it can never be both at the same time. A function like "sqrt" always needs to produce a single result.
