1. **Problem:** Simplify the expression $ (x^{-4})^2 $. 2. **Formula:** When raising a power to another power, multiply the exponents: $$ (a^m)^n = a^{m \times n} $$ 3. **Apply the rule:** $$ (x^{-4})^2 = x^{-4 \times 2} $$ 4. **Calculate the exponent:** $$ x^{-8} $$ 5. **Final answer:** $$ x^{-8} $$ This means the expression simplifies to $x^{-8}$, which can also be written as $\frac{1}{x^8}$ if you want to express it without a negative exponent.