1. Problem: Multiply the radicals and simplify. 2. Formula: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b}$ and $c\sqrt{a} \cdot d\sqrt{b} = (c \times d) \sqrt{a \times b}$. 3. Steps: 1. $\sqrt{5} \cdot \sqrt{3} = \sqrt{15}$ 2. $\sqrt{6} \cdot \sqrt{10} = \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}$ 3. $2\sqrt{7} \cdot 3\sqrt{2} = 6\sqrt{14}$ 4. $4\sqrt{2} \cdot \sqrt{8} = 4\sqrt{16} = 4 \times 4 = 16$ 5. $\sqrt{4} \cdot \sqrt{2} = \sqrt{8} = 2\sqrt{2}$ 6. $5\sqrt{3} \cdot 2\sqrt{5} = 10\sqrt{15}$ 7. $\sqrt{12} \cdot \sqrt{3} = \sqrt{36} = 6$ 8. $2\sqrt{6} \cdot 4\sqrt{5} = 8\sqrt{30}$ 9. $\sqrt{2} \cdot \sqrt{32} = \sqrt{64} = 8$ 10. $3\sqrt{10} \cdot 2\sqrt{2} = 6\sqrt{20} = 6 \times 2\sqrt{5} = 12\sqrt{5}$ 11. $7\sqrt{2} \cdot \sqrt{18} = 7\sqrt{36} = 7 \times 6 = 42$ 12. $\sqrt{5} \cdot 6\sqrt{10} = 6\sqrt{50} = 6 \times 5\sqrt{2} = 30\sqrt{2}$ 13. $\sqrt{20} \cdot \sqrt{5} = \sqrt{100} = 10$ 14. $2\sqrt{11} \cdot 3\sqrt{3} = 6\sqrt{33}$ 15. $4\sqrt{3} \cdot 2\sqrt{12} = 8\sqrt{36} = 8 \times 6 = 48$ Final answers: 1. $\sqrt{15}$ 2. $2\sqrt{15}$ 3. $6\sqrt{14}$ 4. $16$ 5. $2\sqrt{2}$ 6. $10\sqrt{15}$ 7. $6$ 8. $8\sqrt{30}$ 9. $8$ 10. $12\sqrt{5}$ 11. $42$ 12. $30\sqrt{2}$ 13. $10$ 14. $6\sqrt{33}$ 15. $48$