How to prove Δx and Δp obey the Heisenberg uncertainty principle?, chemistry homework help

The spread in an observable is ΔA =(A^2 – (a)^2)^1/2 where A is the expectation value for the operator A hat.
How do I evaluate knowing this Δx and Δp for the particle in a box to show it obeys Heisenbergs uncertainty principle?
Also, why is the expectation value P^2 not equal to p^2 and the same for the expectation value of X^2 not equal to x^2?

"Is this question part of your assignment? We can help"