Let $A$ be any non-empty set and $\alpha$ be an infinite cardinal . When can we say $|A^\alpha|=\alpha$ ?

Question

Let $A$ be any non-empty set and $\alpha$ be an infinite cardinal . When can we say that the cardinality of $A^\alpha$ ($A \times A\times ...$ $\alpha$ times ) is $\alpha$ ? When $\alpha > |A|$ , can we say $|A^\alpha|=\alpha$ ?

Answer 1

Never.

Either $|A|=1$ in which case $|A^\alpha|=1$, or $|A|>1$ in which case $|A^\alpha|\geq 2^\alpha>\alpha$.