1. The problem is to simplify the expression $$\sqrt{18x^3y^2z}$$ assuming all variables are positive. 2. Recall the property of square roots: $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ and that $$\sqrt{x^2} = x$$ for positive $x$. 3. First, factor the radicand (the expression inside the square root): $$18x^3y^2z = 9 \cdot 2 \cdot x^2 \cdot x \cdot y^2 \cdot z$$ 4. Apply the square root to each factor: $$\sqrt{9 \cdot 2 \cdot x^2 \cdot x \cdot y^2 \cdot z} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{x^2} \cdot \sqrt{x} \cdot \sqrt{y^2} \cdot \sqrt{z}$$ 5. Simplify the perfect squares: $$\sqrt{9} = 3$$ $$\sqrt{x^2} = x$$ $$\sqrt{y^2} = y$$ 6. Substitute back: $$3 \cdot \sqrt{2} \cdot x \cdot \sqrt{x} \cdot y \cdot \sqrt{z}$$ 7. Combine the terms outside and inside the square root: $$3xy \cdot \sqrt{2xz}$$ 8. Therefore, the simplified expression is: $$\boxed{3xy\sqrt{2xz}}$$