Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 16H / ii / Solution

If $\lambda$ is infinite and $2\le\kappa\le\lambda$, then
$$
\kappa^\lambda=2^\lambda.
$$
Indeed,
$$
2^\lambda\le\kappa^\lambda\le\lambda^\lambda
\le(2^\lambda)^\lambda
=2^{\lambda\lambda}=2^\lambda.
$$


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 Incoming links: Cardinal exponentiation between two and the exponent