1. **Problem:** Multiply and simplify the radicals for the first expression: $$\sqrt{6}^3 \cdot \sqrt{8}r^2$$ 2. **Recall:** The product of radicals rule: $$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$ and powers of radicals: $$\sqrt{a}^3 = (\sqrt{a})^3$$. 3. **Rewrite:** $$\sqrt{6}^3 = (\sqrt{6})^3 = (6^{1/2})^3 = 6^{3/2}$$. 4. **Multiply:** $$6^{3/2} \cdot \sqrt{8}r^2 = 6^{3/2} \cdot 8^{1/2} r^2$$. 5. **Combine radicals:** $$6^{3/2} \cdot 8^{1/2} = 6^{3/2} \cdot 8^{1/2} = 6^{1 + 1/2} \cdot 8^{1/2}$$ is incorrect; instead, keep as is and multiply numerically: 6. Calculate each: $$6^{3/2} = 6^{1} \cdot 6^{1/2} = 6 \cdot \sqrt{6}$$ $$8^{1/2} = \sqrt{8} = 2\sqrt{2}$$ 7. Substitute back: $$6^{3/2} \cdot 8^{1/2} = 6 \cdot \sqrt{6} \cdot 2 \sqrt{2} = 12 \cdot \sqrt{6} \cdot \sqrt{2}$$ 8. Multiply radicals: $$\sqrt{6} \cdot \sqrt{2} = \sqrt{12} = 2\sqrt{3}$$ 9. So the product is: $$12 \cdot 2 \sqrt{3} = 24 \sqrt{3}$$ 10. Include the $r^2$ term: $$24 \sqrt{3} r^2$$ **Final answer:** $$24 \sqrt{3} r^2$$