I have been trying to do a product construction(intersection) with 2 DFA which has dead states, but I'm stuck at the transition table with a confusion that,what should I do with dead state. Should I also combine the dead state with other states?
The product construction works for any two DFAs. There is nothing special you need to do. The resulting automaton would have many dead states and so won't be minimal, but that doesn't affect its correctness.