Exercise 4-A of Milnor and Stasheff's book Characteristic Classes reads:
Show that the Stiefel–Whitney classes of a Cartesian product are given by $$w_k(\xi\times\eta) = \sum^k_{i=0} w_i(\xi)\times w_{k-i}(\eta)$$
This does not make sense to me because if $\xi\rightarrow B_1$ and $\eta\rightarrow B_2$ then $\xi\times\eta\rightarrow B_1\times B_2$, so $w_k(\xi\times\eta)\in H^k(B_1\times B_2;\mathbb{Z}_2)$. While $w_i(\xi)\times w_{k-i}(\eta)=(w_i(\xi),w_{k-i}(\eta))\in H^i(B_1;\mathbb{Z}_2)\times H^{k-i}(B_2;\mathbb{Z}_2)$.
So $w_k(\xi\times\eta)$ and $\sum^k_{i=0} w_i(\xi)\times w_{k-i}(\eta)$ lie in different sets, so how can I be asked to prove that they are equal?