1. Multiply under one radical: $$\sqrt{8x} \cdot \sqrt{4x} \cdot \sqrt{3x} = \sqrt{8x \cdot 4x \cdot 3x} = \sqrt{96x^3} = \sqrt{16 \cdot 6 \cdot x^3} = 4x\sqrt{6x}$$ 2. Multiply under one radical: $$3\sqrt{5y} \cdot \sqrt{6y} \cdot \sqrt{12y} = 3\sqrt{5y \cdot 6y \cdot 12y} = 3\sqrt{360y^3} = 3\sqrt{36 \cdot 10 \cdot y^3} = 3 \cdot 6 y \sqrt{10y} = 18 y \sqrt{10y}$$ 3. Multiply coefficients and radicals: $$5\sqrt{3a^2} \cdot 3\sqrt{25a} = 15 \sqrt{3a^2 \cdot 25a} = 15 \sqrt{75 a^3} = 15 \sqrt{25 \cdot 3 \cdot a^3} = 15 \cdot 5 a^{1} \sqrt{3a} = 75 a \sqrt{3a}$$ 4. Multiply coefficients and radicals: $$5\sqrt{3a^3} \cdot 4\sqrt{25a^4} = 20 \sqrt{3a^3 \cdot 25a^4} = 20 \sqrt{75 a^7} = 20 \sqrt{25 \cdot 3 \cdot a^7} = 20 \cdot 5 a^{3} \sqrt{3a} = 100 a^{3} \sqrt{3a}$$ 5. Multiply radicals: $$\sqrt{x + 1} \cdot \sqrt{(x + 1)^4} = \sqrt{(x + 1)^{1 + 4}} = \sqrt{(x + 1)^5} = (x + 1)^{5/2}$$ 6. Multiply radicals: $$\sqrt{a - b} \cdot \sqrt{3(a - b)} = \sqrt{3(a - b)^2} = \sqrt{3} (a - b)$$ 7. Distribute: $$\sqrt{x} (\sqrt{x} + \sqrt{2}) = x + \sqrt{2x}$$ 8. Distribute: $$\sqrt{6} (\sqrt{2} + \sqrt{3}) = \sqrt{12} + \sqrt{18} = 2\sqrt{3} + 3\sqrt{2}$$ 9. Use difference of squares: $$(5 - \sqrt{2})(5 + \sqrt{2}) = 5^2 - (\sqrt{2})^2 = 25 - 2 = 23$$ 10. Simplify inside parentheses: $$(\sqrt{x} - 1 + 2)(\sqrt{x} - 1 - 2) = (\sqrt{x} + 1)(\sqrt{x} - 3) = x - 3\sqrt{x} + \sqrt{x} - 3 = x - 2\sqrt{x} - 3$$ 11. Square binomial: $$(\sqrt{5} + \sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}$$ 12. Multiply: $$(5\sqrt{2} - 3\sqrt{3})(5\sqrt{2} + 3\sqrt{2}) = 25 \cdot 2 + 15 \sqrt{6} - 15 \sqrt{6} - 9 \cdot 6 = 50 - 54 = -4$$ 13. Multiply: $$(4\sqrt{2} + 2\sqrt{3})(\sqrt{3} - 3\sqrt{2}) = 4\sqrt{6} - 12 \cdot 2 + 2 \cdot 3 - 6 \sqrt{6} = 4\sqrt{6} - 24 + 6 - 6\sqrt{6} = -18 - 2\sqrt{6}$$ 14. Multiply: $$(2\sqrt{3} + 4)(2\sqrt{3} - 3) = 4 \cdot 3 - 6 \sqrt{3} + 8 \sqrt{3} - 12 = 12 + 2 \sqrt{3} - 12 = 2 \sqrt{3}$$ 15. Square binomial: $$(\sqrt{5} - \sqrt{3})^2 = 5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}$$ 16. Square binomial: $$(3\sqrt{4} - \sqrt{2})^2 = (6 - \sqrt{2})^2 = 36 - 12\sqrt{2} + 2 = 38 - 12\sqrt{2}$$ 17. Multiply radicals: $$\sqrt{3} \cdot \sqrt[3]{2} = 3^{1/2} \cdot 2^{1/3}$$ (cannot simplify further) 18. Multiply radicals: $$\sqrt[3]{3} \cdot \sqrt[4]{2} = 3^{1/3} \cdot 2^{1/4}$$ (cannot simplify further) 19. Multiply radicals: $$\sqrt{5} \cdot \sqrt[3]{2} = 5^{1/2} \cdot 2^{1/3}$$ (cannot simplify further) 20. Multiply radicals: $$\sqrt[3]{10} \cdot \sqrt{5} = 10^{1/3} \cdot 5^{1/2}$$ (cannot simplify further) 21. Multiply radicals: $$\sqrt{2} \cdot \sqrt[3]{2} = 2^{1/2} \cdot 2^{1/3} = 2^{(1/2 + 1/3)} = 2^{5/6}$$ 22. Multiply radicals: $$\sqrt{2} \cdot \sqrt[4]{2} = 2^{1/2} \cdot 2^{1/4} = 2^{(1/2 + 1/4)} = 2^{3/4}$$