1. **State the problem:** Simplify the expression $$3\sqrt{8x^{2}y} + 5x\sqrt{2xy^{2}} - x\sqrt{32y}$$. 2. **Recall the rule:** $$\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$$ and $$\sqrt{a^{2}} = a$$ for $a \geq 0$. 3. Simplify each radical term: - $$3\sqrt{8x^{2}y} = 3\sqrt{4 \cdot 2 \cdot x^{2} \cdot y} = 3 \cdot \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{x^{2}} \cdot \sqrt{y} = 3 \cdot 2 \cdot \sqrt{2} \cdot x \cdot \sqrt{y} = 6x\sqrt{2y}$$ - $$5x\sqrt{2xy^{2}} = 5x \cdot \sqrt{2} \cdot \sqrt{x} \cdot \sqrt{y^{2}} = 5x \cdot \sqrt{2} \cdot \sqrt{x} \cdot y = 5xy\sqrt{2x}$$ - $$x\sqrt{32y} = x \cdot \sqrt{16 \cdot 2 \cdot y} = x \cdot \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{y} = x \cdot 4 \cdot \sqrt{2} \cdot \sqrt{y} = 4x\sqrt{2y}$$ 4. Substitute back: $$6x\sqrt{2y} + 5xy\sqrt{2x} - 4x\sqrt{2y}$$ 5. Combine like terms where possible: $$6x\sqrt{2y} - 4x\sqrt{2y} = (6x - 4x)\sqrt{2y} = 2x\sqrt{2y}$$ 6. Final simplified expression: $$2x\sqrt{2y} + 5xy\sqrt{2x}$$