How to prove we must insert parentheses(brackets) with powers Question:
When we want to add three numbers, say $a + b + c$, we don’t bother
inserting parentheses because $(a + b) + c = a + (b + c)$. But with powers, this is not true -
${(a^b)}^c$ need not be equal to $a^{(b^c)}$ - so we must be careful.
Show that this really is a problem,
by finding positive integers $a,b,c$ such that
${(a^b)}^c < a^{(b^c)}$
and positive integers $d,e,f$ such that
${(d^e)}^f > d^{(e^f)}$.

Do I just have to show one actual example of each as an answer or should I write a proof without examples?
If I'm supposed to write a proof, where do I start?
And what would the full proof be?
Thank you so much!
 A: There are definitions behind.
The addition of three numbers is left-associative, i.e., $a+b+c := (a+b)+c$. One can show that it is also right-associative, i.e. it is associative: $(a+b)+c = a+(b+c)$.
The exponentiation of three number is right-associative, i.e., $a^{b^c} =a^{(b^c)}$. As you recognized, it is not left-associative.
A: You don't even need two examples. You only need one example of values $a,b,c$ for which $$a^{(b^c)}\neq \left(a^b\right)^c$$ to prove your point.
Finding any three such numbers is enough to prove that the statement
$$\forall a,b,c: a^{(b^c)}=\left(a^b\right)^c$$ is false, and thus justify that parentheses in this case are necessary.
A: The question is not only to prove that ${(a^b)}^c$ need not be equal to $a^{(b^c)}$, but that no inequality holds in general between these two expressions. In logical terms, the question is to show that neither of the two following formulas is true
\begin{align}
\text{for all } (a, b, c) \in {\Bbb Z}^3 \quad &{(a^b)}^c \leqslant a^{(b^c)}\\
\text{for all } (a, b, c) \in {\Bbb Z}^3 \quad &{(a^b)}^c \geqslant a^{(b^c)}
\end{align}
To prove this, you need to exhibit a counterexample for each of them, that is\begin{align}
\text{there exists } (a, b, c) \in {\Bbb Z}^3 \quad &{(a^b)}^c > a^{(b^c)}\\
\text{there exists } (a, b, c) \in {\Bbb Z}^3 \quad &{(a^b)}^c < a^{(b^c)}
\end{align}
which is exactly what you are asked to do.
