\( \left. \begin{array} { l } { 8 a ^ { 6 } b ^ { - 2 } } \\ { 27 a ^ { - 3 } b } \end{array} \right) ^ { - \frac { 2 } { 3 } } \)

To simplify the expression \( \left( \frac{8a^6b^{-2}}{27a^{-3}b} \right)^{-\frac{2}{3}} \), first, combine the terms inside the parentheses. This results in \( \frac{8a^{6 + 3}b^{-2 - 1}}{27} = \frac{8a^9b^{-3}}{27} \). Next, apply the negative exponent, which means taking the reciprocal, turning it into \( \left( \frac{27}{8a^9b^{-3}} \right)^{\frac{2}{3}} \). Now, apply the exponent \( \frac{2}{3} \) to each component: \( = \frac{27^{\frac{2}{3}}}{8^{\frac{2}{3}}(a^{9})^{\frac{2}{3}}(b^{-3})^{\frac{2}{3}}} \). This results in \( = \frac{9}{4 a^{6} b^{-2}} \). Lastly, express \( b^{-2} \) as \( \frac{1}{b^2} \), yielding \( \frac{9b^2}{4a^6} \) as the final answer.