Why isn't $2^{2^{-1}}$ equivalent to $2^{-2}$?

Question

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?

Answer 1

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.

Answer 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:

$$a^{b^c}=a^{(b^c)}\ne a^{bc}$$

Answer 3

$2^{2^{-1}}=2^{\frac{1}{2}}=\sqrt2$, but $2^{-2}=\frac{1}{4}$.