I was reading Fulton's Intersection Theory and came across this useful formula for the chern class of the tensor product: $$ c_r(E\otimes L) = \sum_{i=0}^r c_1(L)^ic_{r-i}(E). $$ However, I couldn't find a definition of what exactly the product of two chern classes was. Is it purely formal or can it be simplifed?
In particular, for a line bundle $\mathcal{L}=\mathcal{O}(D)$ on a variety $X$, we know $c_1(\mathcal{L}) = D$. It's tempting to write $$ c_1(\mathcal{L})^2 = D^2 = D.D $$ where $D.D$ is the self-intersection as usual, but I can't really justify this from the axioms or basic results I've found.
Is the above true? If so, how does one justify it, and if not, how should I think about products and power of (first) chern classes?
Thanks.