inverse of stiefel-whitney class of product bundles

Question

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a \xi_{j})={w}(\xi_1)\times \cdots\times {w}(\xi_n). $$ Can we write

$$ \bar w(\Pi_{j=1}^a \xi_{j})=\bar{w}(\xi_1)\times \cdots\times \bar{w}(\xi_n)? $$
why?

Answer 1

Sure: $w(\xi_1) \cdots w(\xi_n) \overline{w}(\xi_1) \cdots \overline{w}(\xi_n) = w(\xi_1) \overline{w}(\xi_1) \cdots w(\xi_n)\overline{w}(\xi_n) = 1\in H^*(M, \mathbb{Z}_2)$.