1. **State the problem:** Simplify the expression $$\left(\frac{3x^4}{y}\right)^{-3}$$. 2. **Recall the rule for negative exponents:** For any nonzero expression $a$, $$a^{-n} = \frac{1}{a^n}$$. 3. **Apply the negative exponent rule:** $$\left(\frac{3x^4}{y}\right)^{-3} = \frac{1}{\left(\frac{3x^4}{y}\right)^3}$$ 4. **Rewrite the denominator by raising numerator and denominator to the power 3:** $$\frac{1}{\frac{(3x^4)^3}{y^3}} = \frac{1}{\frac{3^3 (x^4)^3}{y^3}}$$ 5. **Simplify powers:** $$(x^4)^3 = x^{4 \times 3} = x^{12}$$ 6. **Substitute back:** $$\frac{1}{\frac{27 x^{12}}{y^3}}$$ 7. **Divide by a fraction by multiplying by its reciprocal:** $$1 \times \frac{y^3}{27 x^{12}} = \frac{y^3}{27 x^{12}}$$ **Final answer:** $$\boxed{\frac{y^3}{27 x^{12}}}$$