The <Hausdorff formula for cardinal exponentiation> states that for infinite cardinals $\kappa$ and $\lambda$,
$$
(\kappa^+)^\lambda
=\kappa^\lambda\cdot\kappa^+
=\max\{\kappa^\lambda,\kappa^+\}.
$$
Equivalently,
$$
\aleph_{\alpha+1}^{\aleph_\beta}
=\aleph_\alpha^{\aleph_\beta}\cdot\aleph_{\alpha+1}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 128 / 3 / a / i / Solution

{c}
{wiki=Cardinal_number#Cardinal_exponentiation}

For infinite cardinals $\kappa,\lambda$, Hausdorff's formula is
$$
(\kappa^+)^\lambda=\kappa^\lambda\cdot\kappa^+
=\max\{\kappa^\lambda,\kappa^+\}.
$$


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 Incoming links: Hausdorff formula for cardinal exponentiation