$$\sqrt[3]{\frac{a^2}{b}}·\sqrt[4]{\frac{3b^3}{7a^3}}·\sqrt[6]{\frac{7a}{3b^3}}$$

$$=\sqrt[4·3]{{\left(\frac{a^2}{b}\right)}^4}·\sqrt[3·4]{{\left(\frac{3b^3}{7a^3}\right)}^3}·\sqrt[2·6]{{\left(\frac{7a}{3b^3}\right)}^2}$$

$$=\sqrt[12]{\frac{a^8·3^3·b^9·7^2·a^2}{b^4·7^3·a^9·3^2·b^6}}$$

$$=\sqrt[12]{\frac{3^3·7^2·a^8·a^2·b^9}{3^2·7^3·a^9·b^{10}}}$$

$$=\sqrt[12]{\frac{3^3·7^2·a^{10}·b^9}{3^2·7^3·a^9·b^{10}}}$$

$$=\sqrt[12]{3^{3-2}·7^{2-3}·a^{10-9}·b^{9-10}}$$

$$=\sqrt[12]{3·7^{-1}·a·b^{-1}}$$

$$=\sqrt[12]{\frac{3a}{7b}}$$