$\begin{align}  & \text{The general term of expansion for }{{\left( {{x}^{2}}-\frac{2}{3x} \right)}^{9}}\text{will be} \\  & \text{of the form}{{\text{ }}^{9}}{{C}_{r}}{{({{x}^{2}})}^{9-r}}{{\left( -\frac{2}{3x} \right)}^{r}} \\  & {{=}^{9}}{{C}_{r}}({{x}^{18-2r}}){{(-1)}^{r}}{{\left( \frac{2}{3} \right)}^{r}}{{\left( {{x}^{-1}} \right)}^{r}} \\  & {{=}^{9}}{{C}_{r}}{{(-1)}^{r}}{{\left( \frac{2}{3} \right)}^{r}}({{x}^{18-2r}}){{x}^{-r}} \\  & {{=}^{9}}{{C}_{r}}{{(-1)}^{r}}{{\left( \frac{2}{3} \right)}^{r}}{{x}^{18-3r}} \\  & \text{The term will be independent of }x\text{ when }{{x}^{18-3r}}={{x}^{0}} \\  & 18-3r=0 \\  & r=6 \\  & \text{The term will therefore be}{{\text{ }}^{9}}{{C}_{6}}{{(-1)}^{6}}{{\left( \frac{2}{3} \right)}^{6}}{{x}^{18-3(6)}} \\  & =84\times \frac{64}{729}{{x}^{18-18}}=\frac{1792}{243} \\ \end{align}$