I need help for solving Ex. 7C from 'Characteristic Classes' by Milnor/Stasheff: The exercise asks to find a formula for the (total) Stiefel-Whitney class of $\xi^m\otimes\eta^n$ over a paracompact base space in terms of the the SW classes of $\xi^m$ and $\eta^n$. (superscript for dimension of the vector bundle). The book tells me to start off by
computing it for $n=m=1$ as $w_1(\xi^1\otimes\eta^1)=w_1(\xi^1)+w_1(\eta^1)$, where I'm already struggling. My guess would be to use the uniqueness of SW classes over paracompact spaces, but I don't know how to exactly
Compute cohomology of $G_m\times G_n$ by the Künneth formula. This is more or less clear, but I don't understand the role that it plays for the whole exercise
As hint: We can first compute the formula in the special case that both bundles are Whitney sums of line bundles
- Establish the formula $w(\xi^m\otimes\eta^n)=p_{m,n}(w_1(\xi^m),\dots,w_m(\xi^m),w_1(\eta^n),\dots,w_n(\eta^n))$ with $p_{m,n}$ being characterized as $p_{m,n}(\sigma_1,\dots,\sigma_m,\tau_1,\dots,\tau_n)=\prod_i \prod_j (1+s_i+t_j)$, where $\sigma_i$ are the elementary symmetric functions associated to $s_1,\dots,s_m$ and likewise $\tau_j$ and $t_1\dots,t_n$
Can someone please provide stepwise guidance as to how to solve this problem, i.e more or less take me by the hand.