1. **State the problem:** Simplify the expression $$(2u^{-2}v^{2})^{2} \cdot 2v^{-5}$$. 2. **Apply the power of a product rule:** $$(ab)^n = a^n b^n$$, so $$(2u^{-2}v^{2})^{2} = 2^{2} (u^{-2})^{2} (v^{2})^{2}$$. 3. **Simplify each term:** $$2^{2} = 4$$ $$(u^{-2})^{2} = u^{-4}$$ $$(v^{2})^{2} = v^{4}$$ 4. **Rewrite the expression:** $$4 u^{-4} v^{4} \cdot 2 v^{-5}$$ 5. **Multiply constants and like bases:** $$4 \cdot 2 = 8$$ $$v^{4} \cdot v^{-5} = v^{4 + (-5)} = v^{-1}$$ 6. **Final simplified expression:** $$8 u^{-4} v^{-1}$$ 7. **Optional: Write with positive exponents:** $$\frac{8}{u^{4} v}$$