1. **State the problem:** Simplify the expression $$\frac{(16)^{-2}}{(4^{-1})^2}$$. 2. **Recall the exponent rules:** - For any nonzero number $a$ and integers $m,n$, $a^{-m} = \frac{1}{a^m}$. - Power of a power: $(a^m)^n = a^{mn}$. - Division of powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$. 3. **Simplify numerator:** $$16^{-2} = \frac{1}{16^2} = \frac{1}{256}$$ 4. **Simplify denominator:** $$(4^{-1})^2 = 4^{-2} = \frac{1}{4^2} = \frac{1}{16}$$ 5. **Rewrite the expression:** $$\frac{\frac{1}{256}}{\frac{1}{16}}$$ 6. **Divide fractions by multiplying numerator by reciprocal of denominator:** $$\frac{1}{256} \times \frac{16}{1} = \frac{16}{256}$$ 7. **Simplify the fraction:** $$\frac{16}{256} = \frac{\cancel{16}^1}{\cancel{16}^{16}} = \frac{1}{16}$$ **Final answer:** $$\frac{1}{16}$$