Boolean Simplification of a large problem

Question

I am unsure where to even start on this problem. My intuition that what ever can be done to the original problem can be done over and over to simplify the whole thing. Please help with some guidance.

$ (a \cdot b \cdot c \cdot d) + (a \cdot b \cdot c \cdot \overline{d}) + (a \cdot b \cdot \overline{c} \cdot d) + (a \cdot b \cdot \overline{c} \cdot \overline{d}) + (a \cdot \overline{b} \cdot c \cdot d) + (a \cdot \overline{b} \cdot c \cdot \overline{d}) + (a \cdot \overline{b} \cdot d) $

Answer 1

HINT: $(a\cdot b\cdot c\cdot d)+(a\cdot b\cdot c\cdot \overline d)=a\cdot b\cdot c\cdot(d+\overline d)=a\cdot b\cdot c$. There are several other places where you can apply that same idea.

Answer 2

Hint: I guess $\cdot$ denotes conjunction [AND] and $+$ disjunction [OR]. Use the distributivity laws $$\begin{aligned} X\cdot Y+X\cdot Z&\,=\,X\cdot(Y+Z)\\ (X+Y)\cdot(X+Z)&\,=\,X+(Y\cdot Z)\,. \end{aligned}$$