[tex]\left(\dfrac{16}9\right)^{2/3} \cdot \left(\dfrac43\right)^{5/3} = \left(\left(\dfrac{16}9\right)^2\right)^{1/3} \cdot \left(\left(\dfrac43\right)^5\right)^{1/3} \\\\ \cdots = \left(\left(\dfrac{16}9\right)^2 \cdot \left(\dfrac43\right)^5\right)^{1/3} \\\\ \cdots = \left(\left(\dfrac{4^2}{3^2}\right)^2 \cdot \left(\dfrac43\right)^5\right)^{1/3} \\\\ \cdots = \left(\left(\left(\dfrac43\right)^2\right)^2 \cdot \left(\dfrac43\right)^5\right)^{1/3} \\\\ \cdots = \left(\left(\dfrac43\right)^4 \cdot \left(\dfrac43\right)^5\right)^{1/3} \\\\ \cdots = \left(\left(\dfrac43\right)^9\right)^{1/3} \\\\ \cdots = \left(\dfrac43\right)^{9/3} \\\\ \cdots = \left(\dfrac43\right)^3 \\\\ \cdots = \dfrac{4^3}{3^3} \\\\ \cdots = \boxed{\dfrac{64}{27}}[/tex]
