$$\sqrt{4xy}·\sqrt[6]{x^2{y}^3}·\sqrt[6]{x^9{y}^8}$$

$$=2\sqrt{xy}·\sqrt[6]{x^2{y}^3{x}^9{y}^8}$$

$$=2\sqrt{xy}·\sqrt[6]{x^{11}{y}^{11}}$$

$$=2·\sqrt[3·2]{{\left(xy\right)}^3}·\sqrt[6]{x^{11}{y}^{11}}$$

$$=2·\sqrt[6]{x^3{y}^3}·\sqrt[6]{x^{11}{y}^{11}}$$

$$=2\sqrt[6]{x^{14}{y}^{14}}$$

$$=2\sqrt[6]{x^{12}{y}^{12}·x^2{y}^2}$$

$$=2\sqrt[6]{{\left(x^2{y}^2\right)}^6·x^2{y}^2}$$

$$=2\sqrt[6]{{\left(x^2{y}^2\right)}^6}·\sqrt[6]{x^2{y}^2}$$

$$=2x^2{y}^2·\sqrt[6]{x^2{y}^2}$$

$$=2x^2{y}^2·\sqrt[3·2]{{\left(xy\right)}^2}$$