1. **State the problem:** Simplify the expression $$\frac{x^4}{9y^3} \times \frac{25y^8}{4x^2} \times x^3 y^3 \times \frac{x^4 y^7}{27}$$. 2. **Write the expression clearly:** $$\frac{x^4}{9y^3} \times \frac{25y^8}{4x^2} \times x^3 y^3 \times \frac{x^4 y^7}{27}$$ 3. **Multiply numerators and denominators separately:** Numerator: $$x^4 \times 25y^8 \times x^3 y^3 \times x^4 y^7 = 25 x^{4+3+4} y^{8+3+7} = 25 x^{11} y^{18}$$ Denominator: $$9 y^3 \times 4 x^2 \times 1 \times 27 = 9 \times 4 \times 27 \times x^2 y^3 = 972 x^2 y^3$$ 4. **Combine the fraction:** $$\frac{25 x^{11} y^{18}}{972 x^2 y^3}$$ 5. **Simplify powers by subtracting exponents:** $$= \frac{25}{972} x^{11-2} y^{18-3} = \frac{25}{972} x^9 y^{15}$$ 6. **Final simplified expression:** $$\boxed{\frac{25}{972} x^9 y^{15}}$$ This is the fully simplified form of the given expression.