Is there cardinal number $\kappa$, such that $\kappa$ is not inaccessible and $2^{<\kappa}<\kappa^{<\kappa}$?

Question

Is there cardinal number $\kappa$, such that $\kappa$ is not inaccessible and $\kappa$ such that $2^{<\kappa}<\kappa^{<\kappa}$?

It is trivial that if $\kappa$ is strongly inaccessible then above inequality holds. But can it be true when we do not assume existence of such numbers?

Answer 1

Actually, for a strongly inaccessible cardinal there is equality, and not inequality between the two.

But take any singular strong limit, e.g. $\beth_\omega$, and it is easy to see that the inequality holds.