Why $2^\kappa=\kappa^{\operatorname{cf}{\kappa}}$, if $\kappa$ is a strong limit cardinal?

Question

On Page 58, Set Theory, Thomas Jech(2006) states the following fact without details.

Another fact worth mentioning is: If $\kappa$ is a strong limit cardinal, $2^\kappa=\kappa^{\operatorname{cf}{\kappa}}$.

My question is how to derive this "fact"?

My first guess is that strong limit cardinals are regular, but it's obviously wrong, since if so, definition of inaccessible cardinal would be redundant.

So another question is what can we say about $\operatorname{cf}{\kappa}$ of a strong limit cardinal $\kappa$?

Answer 1

Let me quote the relevant part of a previously proved theorem:

Theorem 5.16.

$(\rm iii)$ If $\kappa$ is a limit cardinal then $2^{\kappa}=(2^{<\kappa})^{\operatorname{cf}(\kappa)}$

By the fact that $\kappa$ is a strong limit we have that $2^{<\kappa}=\sum_{\lambda<\kappa} 2^{\lambda} = \sup\{2^\lambda\mid\lambda<\kappa\}=\kappa$ and so: $$\kappa^{\operatorname{cf}(\kappa)}\leq\kappa^\kappa=2^\kappa=(2^{<\kappa})^{\operatorname{cf}(\kappa)}=\kappa^{\operatorname{cf}(\kappa)}$$