Let $k$ be a field of characteristic 0. Let us consider two elements $A,B \in$ Br$(k)$, which are not inverse to each other.
What is the index of the product $A \otimes B$ in Br$(k)$?
I assume that since the index of an algebra is just the degree of its underlying division algebra, that the index of the product above is just the product of the indexes of $A, B$.
Maybe this is too obvious to be pointed out in any book i found.
The period is a whole other story of course. No: I did no confuse the index with the period.