Absoluteness of exponential function and forcing

Question

I recall reading that the exponential function $ \alpha^{\beta}$ is absolute for transitive models of ZFC.

Is it true that if we have $ 2^{ \alpha } < \beta $ in the ground model $V$, then $( 2^{ \check{ \alpha} } < \check{\beta})^{V[G]} $ holds as well?

Does it make sense to refer to $ 2^{\check{\alpha}}$ as an ordinal in an extension model statement as I did above?

Answer 1

Ordinal arithmetic is not changed between transitive models. And forcing does not change the ordinals.

So $2^\alpha$ is just the same ordinal as it was before.