1. **State the problem:** Multiply the expressions $$\sqrt{4n^3} \cdot \sqrt{50n}$$ assuming all variables represent positive real numbers. 2. **Use the property of radicals:** $$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$$ for positive real numbers. 3. **Apply the property:** $$\sqrt{4n^3} \cdot \sqrt{50n} = \sqrt{4n^3 \cdot 50n}$$ 4. **Multiply inside the radical:** $$4n^3 \cdot 50n = 200n^{4}$$ 5. **Rewrite the expression:** $$\sqrt{200n^{4}}$$ 6. **Simplify the radical:** Since $$n^{4} = (n^{2})^{2}$$, we can take $$n^{2}$$ out of the square root. 7. **Factor 200 to simplify:** $$200 = 100 \times 2$$, and $$\sqrt{100} = 10$$. 8. **Simplify step-by-step:** $$\sqrt{200n^{4}} = \sqrt{100 \times 2 \times n^{4}} = \sqrt{100} \cdot \sqrt{2} \cdot \sqrt{n^{4}} = 10 \cdot \sqrt{2} \cdot n^{2}$$ 9. **Final simplified answer:** $$10 n^{2} \sqrt{2}$$