1. **State the problem:** Simplify the expression $$\left(\frac{9x^{11}}{3x^3}\right)^{-4}$$. 2. **Use the quotient rule for exponents:** When dividing like bases, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Simplify inside the parentheses:** $$\frac{9x^{11}}{3x^3} = \frac{9}{3} \cdot \frac{x^{11}}{x^3} = 3 \cdot x^{11-3} = 3x^8$$. 4. **Rewrite the expression:** $$\left(3x^8\right)^{-4}$$. 5. **Apply the negative exponent rule:** $$a^{-n} = \frac{1}{a^n}$$, so $$\left(3x^8\right)^{-4} = \frac{1}{\left(3x^8\right)^4}$$. 6. **Apply the power of a product rule:** $$\left(ab\right)^n = a^n b^n$$, so $$\frac{1}{3^4 \cdot (x^8)^4}$$. 7. **Simplify powers:** $$3^4 = 81$$ and $$ (x^8)^4 = x^{8 \times 4} = x^{32}$$. 8. **Final simplified expression:** $$\frac{1}{81x^{32}}$$. **Answer:** $$\boxed{\frac{1}{81x^{32}}}$$