The question states:
Simplify $2^{\sqrt2}$
I tried doing: $$2^{\sqrt2}=2^{(2^{\frac{1}{2}})}$$ $$=(2^2)^{\frac{1}{2}}$$ $$=4^{\frac{1}{2}}$$ $$=2$$ $$=2^1$$
Clearly this is ridiculous as I've just "proved" $2^{\sqrt2}=2^1$. Yet my proof seems quite convincing to me. Where did I go wrong? I tried debugging by trying a similar technique on $2^4$ and it worked fine.
$$2^{4}=2^{2^{2}}$$ $$=(2^2)^{2}$$ $$=4^{2}$$ $$=16$$ $$=2^4$$
Because $x^{(y^z)}$ (like you did in the first example) does not mean $(x^y)^z$. It works in your second example because by chance $2^2=2\times 2$.
We say that the exponentiation does not have the associative property.