1. **State the problem:** Simplify the expression $$\sqrt{32x^4} \cdot \sqrt{50x^2} \cdot \sqrt{2x^3}$$ so that the radicand has no perfect squares except for 1. 2. **Recall the property of square roots:** $$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$$. We can combine the roots: $$\sqrt{32x^4} \cdot \sqrt{50x^2} \cdot \sqrt{2x^3} = \sqrt{32x^4 \cdot 50x^2 \cdot 2x^3}$$ 3. **Multiply the radicands:** $$32 \cdot 50 \cdot 2 = 3200$$ $$x^4 \cdot x^2 \cdot x^3 = x^{4+2+3} = x^9$$ So the expression becomes: $$\sqrt{3200x^9}$$ 4. **Factor 3200 to extract perfect squares:** $$3200 = 64 \times 50$$ Since $$64 = 8^2$$ is a perfect square, rewrite: $$\sqrt{64 \times 50 \times x^9} = \sqrt{64} \cdot \sqrt{50} \cdot \sqrt{x^9}$$ 5. **Simplify each root:** $$\sqrt{64} = 8$$ $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$ $$\sqrt{x^9} = x^{\frac{9}{2}} = x^4 \cdot \sqrt{x}$$ (since $$x^9 = x^{8} \cdot x^1$$ and $$\sqrt{x^8} = x^4$$) 6. **Combine all parts:** $$8 \times 5\sqrt{2} \times x^4 \sqrt{x} = 40 x^4 \sqrt{2x}$$ **Final simplified expression:** $$40 x^4 \sqrt{2x}$$