Yes. A fairly general answer to such questions is provided by Easton’s theorem: if the ground model satisfies GCH and $F$ is a class function from a subclass of regular cardinals to cardinals such that $F$ is nondecreasing and $\kappa< \operatorname{cf}(F(\kappa))$ for each $\kappa\in\operatorname{dom}(F)$, then one can find construct a forcing extension with the same cardinals and cofinalities which satisfies $2^\kappa=F(\kappa)$ for each $\kappa\in\operatorname{dom}(F)$.
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Yes. A fairly general answer to such questions is provided by Easton’s theorem: if $F$ is a class function from a subclass of regular cardinals to cardinals such that $F$ is nondecreasing and $\kappa< \operatorname{cf}(F(\kappa))$ for each $\kappa\in\operatorname{dom}(F)$, then one can find a forcing extension with the same cardinals and cofinalities which satisfies $2^\kappa=F(\kappa)$ for each $\kappa\in\operatorname{dom}(F)$. |
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