Answer
$s(t)=\frac{t^3}{3}-2t\sqrt{t}+\frac{8}{3}$
Work Step by Step
Note that $s(t)=\int v(t) \ dt \Rightarrow s(t)= \int t^2-3t^{\frac{1}{2}} dt.$
$$=t^3-2t\sqrt{t}+C.$$ since $s(4)=8$, then
$$8=\frac{64}{3}-16 +C \Rightarrow C=\frac{8}{3}$$.