4. \( \frac{\left(6 c d^{-1} e^{2}\right)^{-3}}{2 c^{-11} d^{4}} \)

To simplify the expression \( \frac{(6 c d^{-1} e^{2})^{-3}}{2 c^{-11} d^{4}} \), we first simplify the numerator: 1. Compute \( (6 c d^{-1} e^2)^{-3} = \frac{1}{(6 c d^{-1} e^2)^{3}} = \frac{1}{6^3 c^3 (d^{-1})^3 (e^2)^3} = \frac{1}{216 c^3 d^{-3} e^6} \). By multiplying by \( d^{3} \) we can rewrite this as: \( \frac{d^{3}}{216 c^3 e^6} \). Now, substituting the numerator back into our original fraction, we get: \[ \frac{d^{3}}{216 c^3 e^6} \div (2 c^{-11} d^{4}) = \frac{d^{3}}{216 c^3 e^6} \times \frac{1}{2 c^{-11} d^{4}} = \frac{d^{3}}{216 \cdot 2 c^3 e^6} \cdot \frac{c^{11}}{d^{4}}. \] 2. Combine the \( d \) terms: \[ \frac{d^{3} \cdot c^{11}}{432 c^3 e^6 d^{4}} = \frac{c^{11}}{432 c^3 e^6} \cdot \frac{d^{3}}{d^{4}} = \frac{c^{11}}{432 c^3 e^6 d^{1}}. \] 3. Simplifying further, we remove common factors in \( c^{11} \): \[ = \frac{c^{11-3}}{432 e^{6} d} = \frac{c^{8}}{432 e^{6} d}. \] So, the final simplified expression is: \[ \frac{c^{8}}{432 e^{6} d}. \]