1. **State the problem:** Simplify the expression $$\frac{(4a^1 b^{-1})^{-2}}{4ab^{-4} c^0}$$ and verify if it equals $$\frac{1}{b^6}$$. 2. **Recall the rules:** - Negative exponent rule: $$x^{-n} = \frac{1}{x^n}$$. - Power of a product: $$(xy)^n = x^n y^n$$. - Zero exponent rule: $$x^0 = 1$$. - Division of like bases: $$\frac{x^m}{x^n} = x^{m-n}$$. 3. **Simplify numerator:** $$(4a^1 b^{-1})^{-2} = 4^{-2} (a^1)^{-2} (b^{-1})^{-2} = \frac{1}{4^2} a^{-2} b^{2} = \frac{1}{16} a^{-2} b^{2}$$. 4. **Simplify denominator:** $$4ab^{-4} c^0 = 4 a b^{-4} \times 1 = 4 a b^{-4}$$. 5. **Write the full fraction:** $$\frac{\frac{1}{16} a^{-2} b^{2}}{4 a b^{-4}} = \frac{1}{16} a^{-2} b^{2} \times \frac{1}{4 a b^{-4}} = \frac{1}{16 \times 4} a^{-2} \times \frac{1}{a} b^{2} \times b^{4}$$. 6. **Simplify constants:** $$\frac{1}{16 \times 4} = \frac{1}{64}$$. 7. **Simplify variables:** $$a^{-2} \times a^{-1} = a^{-3}$$ $$b^{2} \times b^{4} = b^{6}$$ 8. **Combine all:** $$\frac{1}{64} a^{-3} b^{6} = \frac{b^{6}}{64 a^{3}}$$. 9. **Final answer:** The simplified expression is $$\frac{b^{6}}{64 a^{3}}$$, which is not equal to $$\frac{1}{b^{6}}$$. Hence, the given expression simplifies to $$\frac{b^{6}}{64 a^{3}}$$, not $$\frac{1}{b^{6}}$$.