1. **State the problem:** Simplify $$\left(4x^{-3}y^{5}\right)^3 \times \left(125x^{-6}y^{12}\right)^{\frac{2}{3}}$$. 2. **Apply the power of a product rule:** $$(ab)^m = a^m b^m$$. 3. Simplify each term separately: $$\left(4x^{-3}y^{5}\right)^3 = 4^3 \times (x^{-3})^3 \times (y^{5})^3 = 64 \times x^{-9} \times y^{15}$$ $$\left(125x^{-6}y^{12}\right)^{\frac{2}{3}} = 125^{\frac{2}{3}} \times (x^{-6})^{\frac{2}{3}} \times (y^{12})^{\frac{2}{3}}$$ 4. Calculate each part: $$125^{\frac{2}{3}} = (5^3)^{\frac{2}{3}} = 5^{3 \times \frac{2}{3}} = 5^2 = 25$$ $$(x^{-6})^{\frac{2}{3}} = x^{-6 \times \frac{2}{3}} = x^{-4}$$ $$(y^{12})^{\frac{2}{3}} = y^{12 \times \frac{2}{3}} = y^{8}$$ So, $$\left(125x^{-6}y^{12}\right)^{\frac{2}{3}} = 25x^{-4}y^{8}$$ 5. Multiply the two results: $$64x^{-9}y^{15} \times 25x^{-4}y^{8} = (64 \times 25) \times x^{-9 + (-4)} \times y^{15 + 8} = 1600x^{-13}y^{23}$$ 6. Express with positive exponents where possible: $$1600 \times \frac{y^{23}}{x^{13}}$$ **Final answer:** $$\boxed{\frac{1600y^{23}}{x^{13}}}$$