Which complexity class $3^{n/3}$
Yes, that is exponential. Something is exponential if it is $2^{O(n^k)}$ for some $k$. You can write $3^{n/3}$ as $2^{n\frac{\log 3}{3 \log 2}}$, so it fits.

It would be more common to write $(3^{1/3})^n \approx 1.44^n$.