How do we know for sure that EXPTIME ≠ P?
The time hierarchy theorem says that, for any reasonable function&nbsp;$f$, there are problems that can be decided in time $O(f(n))$ that cannot be decided in time, say, $O(f(n)/n)$. (There are various versions of the time hierarchy theorem. The one I've just quoted isn't standard but it's strong enough to prove what we need, here.) In particular, then, there are problems that can be solved in time $O(2^n)$ that cannot be solved in time $O(2^n/n)$ so, in particular, cannot be solved in time $O(n^k)$ for any&nbsp;$k$.  These problems are in EXP but not in P.
Correct: we conjecture that P$\,\neq\,$NP and NP$\,\neq\,$EXP but we don't have a proof of either of those things.