1. **State the problem:** Simplify the expression $$(2x^6)^{-2}$$. 2. **Recall the power of a product rule:** When raising a product to a power, apply the exponent to each factor inside the parentheses: $$ (ab)^n = a^n b^n $$ 3. **Apply the rule:** $$ (2x^6)^{-2} = 2^{-2} \cdot (x^6)^{-2} $$ 4. **Simplify each part:** - For the constant: $$ 2^{-2} = \frac{1}{2^2} = \frac{1}{4} $$ - For the variable with exponent: $$ (x^6)^{-2} = x^{6 \times (-2)} = x^{-12} = \frac{1}{x^{12}} $$ 5. **Combine the results:** $$ \frac{1}{4} \cdot \frac{1}{x^{12}} = \frac{1}{4x^{12}} $$ 6. **Final answer:** $$ \boxed{\frac{1}{4x^{12}}} $$ This means the original expression simplifies to the reciprocal of $4x^{12}$.