$[5$ points$]$ Simplify the following expression to Big$-0$ notation, justifying properly using the definition of $O$ notations and$/$or some of its properties.

a$.O(2^n)+O(n^3*logn)$

b$.O(n^3)+O(lo{g}^{2}n)*O(n^{\frac{1}{2}})$

c$.O(logn)*O(2^n)+O(n!)$

d$.O(3^n)+O(n)*O(2^n)$