1. **State the problem:** Simplify the expression $$\frac{2b^{-2} \cdot (2a^{-3} b^{4})^{4}}{a^{-3} b^{2}}$$. 2. **Recall the exponent rules:** - Power of a product: $$(xy)^n = x^n y^n$$ - Power of a power: $$(x^m)^n = x^{mn}$$ - Multiplying powers with the same base: $$x^m \cdot x^n = x^{m+n}$$ - Dividing powers with the same base: $$\frac{x^m}{x^n} = x^{m-n}$$ 3. **Apply the power of a product rule:** $$(2a^{-3} b^{4})^{4} = 2^{4} (a^{-3})^{4} (b^{4})^{4} = 16 a^{-12} b^{16}$$ 4. **Rewrite the numerator:** $$2 b^{-2} \cdot 16 a^{-12} b^{16} = 32 a^{-12} b^{-2 + 16} = 32 a^{-12} b^{14}$$ 5. **Rewrite the denominator:** $$a^{-3} b^{2}$$ 6. **Divide numerator by denominator using exponent subtraction:** $$\frac{32 a^{-12} b^{14}}{a^{-3} b^{2}} = 32 a^{-12 - (-3)} b^{14 - 2} = 32 a^{-9} b^{12}$$ 7. **Final simplified expression:** $$32 a^{-9} b^{12}$$ This is the simplified form of the given expression.