$\nabla\left(\nabla \boldsymbol{r}_{0} e^{i \boldsymbol{k} \boldsymbol{r}} e^{-i \omega t}\right) $- meaning?

Question

My physics professor wrote the following equation: $$ \nabla\left(\nabla \boldsymbol{r}_{0} e^{i \boldsymbol{k r}} e^{-i \omega t}\right)=-e^{-i \omega t} e^{i \boldsymbol{k r}} \boldsymbol{k}\left(\boldsymbol{k} \cdot \boldsymbol{r}_{0}\right) $$ where $\mathbf{k}$ is a general $k$-vector and $r=[r_x,r_y,r_z]^T$ a cartesian vector. Can someone explain how the right-hand side is obtained from the left-hand side? Usually $\nabla$ is used to denote the gradient, but calculating the gradient of the gradient of $\boldsymbol{r}_{0} e^{i \boldsymbol{k r}} e^{-i \omega t}$ does not lead to the expression on the right-hand side.