Cardinal Arithmetic, Regular Cardinals, and Exponentiation

Question

I cannot solve a simple cardinal exponentiation/regularity exercise. Let $\kappa$ be a regular cardinal. Why is the cardinal $\kappa^{\lt \kappa}=\sum_{\alpha<\kappa}\kappa^\alpha$ regular as well? I would appreciate a hint.

Answer 1

This is [consistently] false.

It is consistent that $\aleph_0^{\aleph_0}=\aleph_{\omega_1}$, which is singular. In this case take $\kappa=\aleph_1$ then $$\kappa^{<\kappa}=\aleph_1^{\aleph_0}=2^{\aleph_0}=\aleph_0^{\aleph_0}=\aleph_{\omega_1}.$$

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Of course, this is also consistently true, under $\sf GCH$ we have that $\kappa^{<\kappa}=\kappa$ for every regular cardinal.