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

To simplify the expression: \[ \frac{\left(6 c d^{-1} e^{2}\right)^{-3}}{2 c^{-11} d^{4}} \] **Step 1: Apply the Negative Exponent** First, apply the negative exponent to the numerator: \[ \left(6 c d^{-1} e^{2}\right)^{-3} = 6^{-3} \cdot c^{-3} \cdot d^{3} \cdot e^{-6} \] So the expression becomes: \[ \frac{6^{-3} \cdot c^{-3} \cdot d^{3} \cdot e^{-6}}{2 c^{-11} d^{4}} \] **Step 2: Simplify the Coefficients** Calculate \(6^{-3}\): \[ 6^{-3} = \frac{1}{6^3} = \frac{1}{216} \] So: \[ \frac{\frac{1}{216} \cdot c^{-3} \cdot d^{3} \cdot e^{-6}}{2 c^{-11} d^{4}} = \frac{1}{432} \cdot \frac{c^{-3} \cdot d^{3} \cdot e^{-6}}{c^{-11} d^{4}} \] **Step 3: Simplify the Variables** Combine the exponents for each variable: - For \(c\): \[ c^{-3} \div c^{-11} = c^{-3 - (-11)} = c^{8} \] - For \(d\): \[ d^{3} \div d^{4} = d^{3-4} = d^{-1} \] - For \(e\): \[ e^{-6} \] **Step 4: Combine Everything** Putting it all together: \[ \frac{1}{432} \cdot c^{8} \cdot d^{-1} \cdot e^{-6} = \frac{c^{8}}{432 \cdot d \cdot e^{6}} \] **Final Simplified Form:** \[ \boxed{\dfrac{c^{8}}{432\,d\,e^{6}}} \]