= Solution
<König theorem for cardinal numbers> states that if $\kappa_i<\lambda_i$ for each $i\in I$, then $\sum_i\kappa_i<\prod_i\lambda_i$. Its <cofinality> formulation gives, for every infinite <cardinal> $\kappa$,
$$
\boxed{\kappa^{\operatorname{cf}(\kappa)}>\kappa.}
$$
Taking a cofinal sequence of smaller <cardinals> in $\kappa$ and comparing their sum with the product of their successors gives this inequality. Another standard consequence, for infinite $\lambda$ and $\kappa\ge2$, is
$$
\boxed{\operatorname{cf}(\kappa^\lambda)>\lambda,\qquad
\operatorname{cf}(2^\lambda)>\lambda.}
$$
Indeed, if $\mu=\operatorname{cf}(\kappa^\lambda)\le\lambda$, the preceding inequality at $\kappa^\lambda$ would contradict $(\kappa^\lambda)^\mu=\kappa^{\lambda\mu}=\kappa^\lambda$.
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