15. 6 ab⁶√81a⁵b ⋅ 1/3 a² √3ab ⋅ b³/9ab².

$$ab\sqrt[6]{81a^5b} \cdot \frac{1}{3} a^2 \sqrt{3ab} \cdot \frac{b^3}{\sqrt[9]{ab^2}} = ab \cdot 81^{\frac{1}{6}} a^{\frac{5}{6}} b^{\frac{1}{6}} \cdot \frac{1}{3} a^2 \cdot 3^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}} \cdot \frac{b^3}{a^{\frac{1}{9}} b^{\frac{2}{9}}} = a^{1+\frac{5}{6}+2+\frac{1}{2}-\frac{1}{9}} b^{1+\frac{1}{6}+\frac{1}{2}+3-\frac{2}{9}} \cdot \frac{81^{\frac{1}{6}}}{3} \cdot 3^{\frac{1}{2}} = a^{\frac{54+45+108+27-2}{54}} b^{\frac{54+9+27+162-12}{54}} \cdot 3^{\frac{4}{6}-1+\frac{1}{2}} = a^{\frac{232}{54}} b^{\frac{240}{54}} \cdot 3^{\frac{2-3+3}{3}} = a^{\frac{116}{27}} b^{\frac{40}{9}} \cdot 3^{\frac{2}{3}}$$

Ответ: $$a^{\frac{116}{27}} b^{\frac{40}{9}} \cdot 3^{\frac{2}{3}}$$