I'm trying to find a solution for exercise G1 of Chapter VII in Kunen's &quot;Set Theory - An Introduction to Independence Proofs&quot;. The question is in the context of Cohen forcing. We start with a c.t.m. $M$ for a sufficiently large segment of ZFC. In the ground model we fix three infinite cardinals $\kappa,\lambda,\vartheta$. The notion of forcing is $P := \mathrm{Fn}(\kappa\times\omega, 2)$. One now has to prove that $$(\lambda^\vartheta)^{M[G]} = ((\max\{\lambda, \kappa\})^\vartheta)^M$$ for any generic filter $G\subseteq P$. I'm guessing that the argument uses the fact that $P$ has c.c.c. and since $|P| = \kappa$ there are at most $\kappa^{\aleph_0}$ many antichains in $P$. However, I do not see how to start the argument. Would someone be so kind as to give me a hint?
Let $\kappa, \lambda,\theta$ be infinite cardinals of $M$ where $M$ is a transitive model of ZFC. Let $\mathcal{P}=Fn(\kappa\times \omega,2)$ (finite functions from $\kappa\times \omega$ to 2) and G is a (M,P)-generic set. I want to show $(\lambda^\theta)^{M[G]}=(max(\kappa,\lambda)^\theta)^M$. One direction is easy $(\lambda^\theta)^M \leq (\lambda^\theta)^{M[G]}$ and $(\kappa^\theta)^M\leq (\kappa^\theta)^{M[G]}\leq ((2^\omega)^\theta)^{M[G]}=(2^\theta)^{M[G]}\leq (\lambda^\theta)^{M[G]}$ where $(\kappa \leq 2^\omega)^{M[G]}$ is by the forcing construction. But now I am not quite sure how to get the other direction. Any thoughts?
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