Solution
= Solution
The weights of $V_2$ are $z^2,1,z^{-2}$. Degree-three monomials in these weights show that the character of the <symmetric power> $S^3V_2$ has weights
$$
z^6+z^4+2z^2+2+2z^{-2}+z^{-4}+z^{-6}
=\chi_{V_6}+\chi_{V_2}.
$$
Characters determine finite-dimensional $SU(2)$ representations, hence
$$
\boxed{S^3V_2\cong V_6\oplus V_2}.
$$