1. **State the problem:** Multiply the expressions $$\sqrt{10c^5} \cdot \sqrt{8c}$$ assuming all variables represent positive real numbers. 2. **Recall the property of radicals:** $$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$ for positive $a$ and $b$. 3. **Apply the property:** $$\sqrt{10c^5} \cdot \sqrt{8c} = \sqrt{10c^5 \cdot 8c} = \sqrt{80c^{6}}$$ 4. **Simplify inside the radical:** $$80c^{6} = 16 \times 5 \times c^{6}$$ 5. **Use the property $$\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}$$ to separate perfect squares:** $$\sqrt{80c^{6}} = \sqrt{16} \cdot \sqrt{5} \cdot \sqrt{c^{6}}$$ 6. **Simplify each radical:** - $$\sqrt{16} = 4$$ - $$\sqrt{5}$$ remains as is (since 5 is not a perfect square) - $$\sqrt{c^{6}} = c^{3}$$ because $$\sqrt{c^{6}} = c^{6/2} = c^{3}$$ 7. **Combine all parts:** $$4 \cdot c^{3} \cdot \sqrt{5} = 4c^{3}\sqrt{5}$$ **Final answer:** $$\boxed{4c^{3}\sqrt{5}}$$