1. **Stating the problem:** Simplify the expression $$\frac{(a^2 b^5 c^4)}{(ab)^2}$$ where $$a=\left(\frac{2}{9}\right)^4$$, $$b=\left(\frac{4}{27}\right)^3$$, and $$c=\left(\frac{8}{9}\right)^3$$, and express the result in index form. 2. **Write the expression with given values:** $$\frac{\left(\left(\frac{2}{9}\right)^4\right)^2 \left(\left(\frac{4}{27}\right)^3\right)^5 \left(\left(\frac{8}{9}\right)^3\right)^4}{\left(\left(\frac{2}{9}\right)^4 \left(\frac{4}{27}\right)^3\right)^2}$$ 3. **Apply power of a power rule:** $$\left(x^m\right)^n = x^{mn}$$ Numerator: $$\left(\frac{2}{9}\right)^{4 \times 2} \times \left(\frac{4}{27}\right)^{3 \times 5} \times \left(\frac{8}{9}\right)^{3 \times 4} = \left(\frac{2}{9}\right)^8 \times \left(\frac{4}{27}\right)^{15} \times \left(\frac{8}{9}\right)^{12}$$ Denominator: $$\left(\frac{2}{9}\right)^{4 \times 2} \times \left(\frac{4}{27}\right)^{3 \times 2} = \left(\frac{2}{9}\right)^8 \times \left(\frac{4}{27}\right)^6$$ 4. **Rewrite the expression:** $$\frac{\left(\frac{2}{9}\right)^8 \times \left(\frac{4}{27}\right)^{15} \times \left(\frac{8}{9}\right)^{12}}{\left(\frac{2}{9}\right)^8 \times \left(\frac{4}{27}\right)^6}$$ 5. **Cancel common factors:** $$= \frac{\cancel{\left(\frac{2}{9}\right)^8} \times \left(\frac{4}{27}\right)^{15} \times \left(\frac{8}{9}\right)^{12}}{\cancel{\left(\frac{2}{9}\right)^8} \times \left(\frac{4}{27}\right)^6} = \left(\frac{4}{27}\right)^{15-6} \times \left(\frac{8}{9}\right)^{12} = \left(\frac{4}{27}\right)^9 \times \left(\frac{8}{9}\right)^{12}$$ 6. **Final answer in index form:** $$\boxed{\left(\frac{4}{27}\right)^9 \times \left(\frac{8}{9}\right)^{12}}$$