1. **State the problem:** Simplify the expression $$\frac{\left(\frac{2}{3} a^{9} b^{-10} c^{4}\right)^{-1} \left(\frac{5}{4} a^{-2} b^{3} c^{-5}\right)^{-2}}{\left(-\frac{1}{3} a^{2} b c\right)^{-2}}$$ 2. **Recall the rules:** - Negative exponent rule: $x^{-n} = \frac{1}{x^n}$ - Power of a power: $(x^a)^b = x^{ab}$ - When dividing powers with the same base, subtract exponents: $\frac{x^m}{x^n} = x^{m-n}$ - When multiplying powers with the same base, add exponents: $x^m \cdot x^n = x^{m+n}$ 3. **Simplify each part:** - First term: $$\left(\frac{2}{3} a^{9} b^{-10} c^{4}\right)^{-1} = \frac{1}{\frac{2}{3} a^{9} b^{-10} c^{4}} = \frac{3}{2} a^{-9} b^{10} c^{-4}$$ - Second term: $$\left(\frac{5}{4} a^{-2} b^{3} c^{-5}\right)^{-2} = \left(\frac{4}{5} a^{2} b^{-3} c^{5}\right)^{2} = \left(\frac{4}{5}\right)^2 a^{4} b^{-6} c^{10} = \frac{16}{25} a^{4} b^{-6} c^{10}$$ - Third term (denominator): $$\left(-\frac{1}{3} a^{2} b c\right)^{-2} = \left(-3 a^{-2} b^{-1} c^{-1}\right)^{2} = (-3)^2 a^{-4} b^{-2} c^{-2} = 9 a^{-4} b^{-2} c^{-2}$$ 4. **Combine numerator terms:** $$\frac{3}{2} a^{-9} b^{10} c^{-4} \times \frac{16}{25} a^{4} b^{-6} c^{10} = \frac{3}{2} \times \frac{16}{25} a^{-9+4} b^{10-6} c^{-4+10} = \frac{48}{50} a^{-5} b^{4} c^{6} = \frac{24}{25} a^{-5} b^{4} c^{6}$$ 5. **Divide by denominator:** $$\frac{\frac{24}{25} a^{-5} b^{4} c^{6}}{9 a^{-4} b^{-2} c^{-2}} = \frac{24}{25} \times \frac{1}{9} a^{-5 - (-4)} b^{4 - (-2)} c^{6 - (-2)} = \frac{24}{225} a^{-1} b^{6} c^{8} = \frac{8}{75} a^{-1} b^{6} c^{8}$$ 6. **Rewrite with positive exponents:** $$\frac{8 b^{6} c^{8}}{75 a}$$ **Final answer:** $$\boxed{\frac{8 b^{6} c^{8}}{75 a}}$$