1. **State the problem:** Evaluate the expression $$\frac{(-8)^{-2}}{3^{-4}}$$. 2. **Recall the rule for negative exponents:** For any nonzero number $a$ and integer $n$, $$a^{-n} = \frac{1}{a^n}$$. 3. **Apply the rule to numerator and denominator:** $$(-8)^{-2} = \frac{1}{(-8)^2}$$ $$3^{-4} = \frac{1}{3^4}$$ 4. **Rewrite the original expression using these:** $$\frac{\frac{1}{(-8)^2}}{\frac{1}{3^4}}$$ 5. **Dividing by a fraction is multiplying by its reciprocal:** $$\frac{1}{(-8)^2} \times \frac{3^4}{1} = \frac{3^4}{(-8)^2}$$ 6. **Calculate powers:** $$(-8)^2 = (-8) \times (-8) = 64$$ $$3^4 = 3 \times 3 \times 3 \times 3 = 81$$ 7. **Substitute back:** $$\frac{81}{64}$$ 8. **Final answer:** $$\boxed{\frac{81}{64}}$$