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!