1. **Problem:** Calculate $$\left[\left(\frac{2}{3}\right)^{-4}\right]^{\frac{2}{3}}$$ and $$-1.5 : \left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121^{\frac{5}{8}}$$. **Step 1:** Simplify the first expression. $$\left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^4 = \frac{3^4}{2^4} = \frac{81}{16}$$ **Step 2:** Raise to the power $$\frac{2}{3}$$: $$\left(\frac{81}{16}\right)^{\frac{2}{3}} = \frac{81^{\frac{2}{3}}}{16^{\frac{2}{3}}}$$ **Step 3:** Calculate numerator and denominator separately. $$81^{\frac{2}{3}} = \left(3^4\right)^{\frac{2}{3}} = 3^{\frac{8}{3}} = 3^{2 + \frac{2}{3}} = 3^2 \cdot 3^{\frac{2}{3}} = 9 \cdot 3^{\frac{2}{3}}$$ $$16^{\frac{2}{3}} = \left(2^4\right)^{\frac{2}{3}} = 2^{\frac{8}{3}} = 2^{2 + \frac{2}{3}} = 4 \cdot 2^{\frac{2}{3}}$$ **Step 4:** So, $$\left(\frac{81}{16}\right)^{\frac{2}{3}} = \frac{9 \cdot 3^{\frac{2}{3}}}{4 \cdot 2^{\frac{2}{3}}}$$ **Step 5:** Now calculate the second expression: $$-1.5 : \left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121^{\frac{5}{8}} = \frac{-1.5}{\left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121^{\frac{5}{8}}}$$ **Step 6:** Simplify powers: $$121 = 11^2$$ $$121^{\frac{5}{8}} = \left(11^2\right)^{\frac{5}{8}} = 11^{\frac{10}{8}} = 11^{\frac{5}{4}}$$ **Step 7:** Final expression for the second part: $$\frac{-1.5}{\left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 11^{\frac{5}{4}}}$$ **Answer 1:** $$\left[\left(\frac{2}{3}\right)^{-4}\right]^{\frac{2}{3}} = \frac{9 \cdot 3^{\frac{2}{3}}}{4 \cdot 2^{\frac{2}{3}}}$$ $$-1.5 : \left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 121^{\frac{5}{8}} = \frac{-1.5}{\left(\frac{3}{2}\right)^{\frac{1}{9}} \cdot 11^{\frac{5}{4}}}$$