Let $\kappa < \beth\_2$ and $\lambda<\beth\_1$ be cardinals. What can we say about $\kappa^{\lambda}$ without assuming CH? Is it true that $\kappa^{\lambda} < \beth\_2$ or $\kappa^{<\beth\_1} < \beth\_2$? 



Cardinal exponentiations inequality
Even if CH holds, this can break.


Suppose $\beth\_1=\aleph\_1$ and $\beth\_2=\aleph\_{\omega+1}$. Let $\lambda=\aleph\_0$ and $\kappa=\aleph\_\omega$. By [Konig's theorem](https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_(set_theory)) we know that $\kappa^\lambda=\kappa^{cf(\kappa)}>\kappa$; since $\kappa^+=\beth\_2$ this means $\kappa^\lambda\ge\beth\_2$ (which in turn means $\kappa^\lambda=\beth\_2$, of course).


