1. **State the problem:** Multiply the expressions $$\sqrt{5x^8y^2} \times \sqrt{10x^3} \times \sqrt{12y}$$ assuming $$x \geq 0$$ and $$y \geq 0$$. 2. **Recall the property of square roots:** $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$ for non-negative $$a$$ and $$b$$. 3. **Combine the square roots:** $$\sqrt{5x^8y^2} \times \sqrt{10x^3} \times \sqrt{12y} = \sqrt{5x^8y^2 \times 10x^3 \times 12y}$$ 4. **Multiply inside the radical:** $$5 \times 10 \times 12 = 600$$ $$x^8 \times x^3 = x^{8+3} = x^{11}$$ $$y^2 \times y = y^{2+1} = y^3$$ So, $$\sqrt{600 x^{11} y^3}$$ 5. **Simplify the radical by factoring perfect squares:** $$600 = 100 \times 6$$, and $$100$$ is a perfect square. 6. **Rewrite the expression:** $$\sqrt{100 \times 6 \times x^{11} \times y^3} = \sqrt{100} \times \sqrt{6} \times \sqrt{x^{11}} \times \sqrt{y^3}$$ 7. **Simplify each square root:** $$\sqrt{100} = 10$$ $$\sqrt{x^{11}} = x^{\frac{11}{2}} = x^5 \times x^{\frac{1}{2}} = x^5 \sqrt{x}$$ $$\sqrt{y^3} = y^{\frac{3}{2}} = y \times y^{\frac{1}{2}} = y \sqrt{y}$$ 8. **Combine all parts:** $$10 \times \sqrt{6} \times x^5 \sqrt{x} \times y \sqrt{y} = 10 x^5 y \times \sqrt{6xy}$$ 9. **Final answer:** $$\boxed{10 x^5 y \sqrt{6 x y}}$$ This matches the option: 10x^5 y √6xy.