Let $2 \le \kappa<\lambda$(both cardinal numbers), in which $\lambda$ is infinite. Then these formula as follows hold where in ZFC:
$\lambda+\kappa=\lambda$
$\lambda\cdot\kappa=\lambda$
$\kappa^\lambda=2^\lambda$
However, if $\lambda$ is not Dedekind-infinite, then 1,2 fail.
But for 3, it's not quite clear.
To prove it. Obviously $2^\lambda\le\kappa^\lambda$; for the other direction, I only got $\kappa^\lambda\le 2^{\kappa \cdot \lambda}$, but $2^{\kappa \cdot \lambda}=2^{\lambda}$ seems not valid.
So my question: Is 3 also valid in all set models of ZF?