One of the rules of powers is that you can multiply higher powers to each other: $2^{2^3}=2^6$.
Therefore, $2^{2^{-1}}$ should equal $2^{-2}$?.
But according to wolfram alpha, $2^{2^{-1}} = 2^\frac{1}{2}$.
Do the rules of power change when a negative sign is present? Does taking the inverse take priority over multiplication?
Because $2^{2^{-1}}$ is not $2^{2\cdot(-1)}$, just $2^\frac{1}{2}$.
That is not the correct rule. You must use parenthesis to clarify. As you have learned:
$$(a^b)^c=a^{bc}$$
but as you tried to put into WolframAlpha:
$$a^{b^c}=a^{(b^c)}\ne a^{bc}$$
It depends on where you put the parentheses. In fact $2^{2^3}= 2^8$ is different that $(2^2)^3=2^{6}$. If it is lacking parntheses, you should evaluate from top to bottom.
$2^{2^{-1}}=2^{\frac{1}{2}}=\sqrt2$, but $2^{-2}=\frac{1}{4}$.