1. **State the problem:** Simplify the expression $$6x^3 y \sqrt{9x^3 y^7}$$ assuming $x$ and $y$ are positive. 2. **Recall the rule for square roots:** $$\sqrt{a^2} = a$$ for positive $a$, and $$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$$. 3. **Rewrite the square root:** $$\sqrt{9x^3 y^7} = \sqrt{9} \times \sqrt{x^3} \times \sqrt{y^7}$$ 4. **Simplify each part:** - $$\sqrt{9} = 3$$ - $$\sqrt{x^3} = \sqrt{x^2 \times x} = x \sqrt{x}$$ - $$\sqrt{y^7} = \sqrt{y^6 \times y} = y^3 \sqrt{y}$$ 5. **Combine the simplified parts inside the root:** $$\sqrt{9x^3 y^7} = 3 x y^3 \sqrt{x y}$$ 6. **Multiply by the outside term:** $$6x^3 y \times 3 x y^3 \sqrt{x y} = 18 x^4 y^4 \sqrt{x y}$$ 7. **Final simplified expression:** $$\boxed{18 x^4 y^4 \sqrt{x y}}$$