Determine whether $*$ is associative, where $*$ is defined on $\Bbb Z^+$ by $a*b=2^{ab}$.

Solution 1:

You can easily find a counterexample to these. Just take the numbers $2,3,5$. We have that

\begin{eqnarray*} (2*3)*5 & = & (2^6)*5 \;\; =\;\; 2^{64\times 5} \;\; =\;\; 2^{320} \\ 2*(3*5) & = & 2*(2^{15}) \;\; =\;\; 2^{2 \times 32,768} \;\; =\;\; 2^{65,536} \end{eqnarray*}

which are quite different from one another.

Solution 2:

As you say, $(a*b)*c = 2^{(a*b)c} = 2^{2^{ab}c}$ vs. $a*(b*c) = 2^{a(b*c)} = 2^{a2^{bc}}$. For these two to be equal, then we would need:

$$ c2^{ab} = a2^{bc} $$

These are clearly not equal for all $a$, $b$, $c$. Simply choose $a$ and $c$ to be relatively prime and then there is no way for these to be equal (in fact, I suspect the only way for these to be equal is for $a = c$).