Boolean Algebra with Demorgans Law
!(a(b + z(x + !a)))
= !(ab + azx + az!a)  [distribution]
= !(ab + azx)         [a./a = 0]  
= !(a.(b+zx))         [factoring]
= !a + !(b+zx)        [DeMorgan]
= !a + !b.!(zx)       [DeMorgan]
= !a + !b.(!z + !x)   [DeMorgan]


There are not many rules to learn in Boolean logic.
Get used to them : There is DeMorgan, factoring, things like a+/a.b = a+b, etc...