1. **State the problem:** Simplify the expression $$\left\{\left[\left(\frac{2}{3}\right)^{-4}\right]^{\frac{2}{3}} \cdot 1.5 \cdot \left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121^{\frac{5}{8}}\right\}$$. 2. **Recall exponent rules:** - $\left(a^m\right)^n = a^{mn}$ - $a^{-m} = \frac{1}{a^m}$ - $a^m \cdot a^n = a^{m+n}$ 3. **Simplify inside the brackets:** $$\left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^4$$ 4. **Apply the outer exponent:** $$\left[\left(\frac{2}{3}\right)^{-4}\right]^{\frac{2}{3}} = \left(\frac{3}{2}\right)^{4 \cdot \frac{2}{3}} = \left(\frac{3}{2}\right)^{\frac{8}{3}}$$ 5. **Rewrite the entire expression:** $$\left(\frac{3}{2}\right)^{\frac{8}{3}} \cdot 1.5 \cdot \left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121^{\frac{5}{8}}$$ 6. **Combine powers of $\frac{3}{2}$:** $$\left(\frac{3}{2}\right)^{\frac{8}{3} + \frac{1}{9}} = \left(\frac{3}{2}\right)^{\frac{24}{9} + \frac{1}{9}} = \left(\frac{3}{2}\right)^{\frac{25}{9}}$$ 7. **Convert 1.5 to fraction:** $$1.5 = \frac{3}{2}$$ 8. **Multiply $\frac{3}{2}$ with the combined power:** $$\frac{3}{2} \cdot \left(\frac{3}{2}\right)^{\frac{25}{9}} = \left(\frac{3}{2}\right)^{1 + \frac{25}{9}} = \left(\frac{3}{2}\right)^{\frac{9}{9} + \frac{25}{9}} = \left(\frac{3}{2}\right)^{\frac{34}{9}}$$ 9. **Simplify $121^{\frac{5}{8}}$:** Since $121 = 11^2$, $$121^{\frac{5}{8}} = \left(11^2\right)^{\frac{5}{8}} = 11^{2 \cdot \frac{5}{8}} = 11^{\frac{10}{8}} = 11^{\frac{5}{4}}$$ 10. **Final expression:** $$\left(\frac{3}{2}\right)^{\frac{34}{9}} \cdot 11^{\frac{5}{4}}$$ **Answer:** $$\boxed{\left(\frac{3}{2}\right)^{\frac{34}{9}} \cdot 11^{\frac{5}{4}}}$$