For elementary reactions, rate = $k[A]^2[B_2]$. When volume is reduced by factor 3, all concentrations increase by factor 3: $[A]_{\text{new}} = 3[A]$, $[B_2]_{\text{new}} = 3[B_2]$. New rate $= k(3[A])^2(3[B_2]) = 9\times3\times k[A]^2[B_2] = 27\times$ (original rate). The rate increases by a factor of 27.