$\newcommand{\cf}{\operatorname{cf}}$ Let $\alpha$, $\beta$ be ordinals. I believe that we have \begin{align*} \cf(\alpha+\beta)=\cf(\beta),\quad\beta\neq 0;\quad\cf(\alpha\cdot\beta)=\begin{cases}\cf(\beta),&\text{if }\beta=0\text{ or }\beta\text{ is a limit ordinal};\\ \cf(\alpha),&\text{if }\beta\text{ is a successor ordinal},\end{cases}\quad\alpha\neq 0;\\ \cf(\alpha^\beta)=\begin{cases}\cf(\beta),&\text{if }\beta\text{ is a limit ordinal};\\ \cf(\beta'),&\text{if }\alpha\text{ is a successor ordinal and }\beta=\beta'+n,\text{ where }\beta'\text{ is a limit ordinal};\\ \cf(\alpha),&\text{if }\alpha\text{ is a limit ordinal and }\beta\text{ is a successor ordinal, or if }\beta\text{ is finite},\end{cases}\quad\alpha\neq 0,1\text{ and }\beta\neq 0. \end{align*} In fact, the point is just to show that for limit ordinal $\beta$, we have $$ \cf(\alpha+\beta)=\cf(\alpha\cdot\beta)=\cf(\alpha^\beta)=\cf(\beta). $$ That $\cf(\alpha+\beta),\cf(\alpha\cdot\beta),\cf(\alpha^\beta)\le\cf(\beta)$ is almost by definition of ordinal arithmetic. For $\ge$, we can use the same technique as in the answer here.
As these results seem to be fairly fundamental, I was hoping to find some references that list these properties when introducing the concept of cofinality.
