1. **Problem Statement:** Simplify the following expressions involving multiplication of surds: - (\sqrt{3} - 1)(\sqrt{3} + 1) - (3 - \sqrt{2})(2 + 3\sqrt{2}) - (3\sqrt{5} - \sqrt{3})(2\sqrt{5} - 3\sqrt{3}) - (3\sqrt{18} - 2\sqrt{27})(2\sqrt{32} + \sqrt{48}) 2. **Formula and Rules:** - Use distributive property (FOIL) for multiplication: $ (a+b)(c+d) = ac + ad + bc + bd $ - Simplify surds by factoring out perfect squares: $ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} $ - Combine like terms carefully. 3. **Step-by-step Solutions:** **(a) Simplify $(\sqrt{3} - 1)(\sqrt{3} + 1)$:** $$ (\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 + \sqrt{3} \times 1 - 1 \times \sqrt{3} - 1^2 $$ $$ = 3 + \cancel{\sqrt{3}} - \cancel{\sqrt{3}} - 1 $$ $$ = 3 - 1 = 2 $$ **(b) Simplify $(3 - \sqrt{2})(2 + 3\sqrt{2})$:** $$ = 3 \times 2 + 3 \times 3\sqrt{2} - \sqrt{2} \times 2 - \sqrt{2} \times 3\sqrt{2} $$ $$ = 6 + 9\sqrt{2} - 2\sqrt{2} - 3 \times (\sqrt{2})^2 $$ $$ = 6 + (9\sqrt{2} - 2\sqrt{2}) - 3 \times 2 $$ $$ = 6 + 7\sqrt{2} - 6 $$ $$ = \cancel{6} + 7\sqrt{2} - \cancel{6} = 7\sqrt{2} $$ **(c) Simplify $(3\sqrt{5} - \sqrt{3})(2\sqrt{5} - 3\sqrt{3})$:** $$ = 3\sqrt{5} \times 2\sqrt{5} - 3\sqrt{5} \times 3\sqrt{3} - \sqrt{3} \times 2\sqrt{5} + \sqrt{3} \times 3\sqrt{3} $$ $$ = 6 \times (\sqrt{5})^2 - 9 \sqrt{15} - 2 \sqrt{15} + 3 \times (\sqrt{3})^2 $$ $$ = 6 \times 5 - 11 \sqrt{15} + 3 \times 3 $$ $$ = 30 - 11 \sqrt{15} + 9 $$ $$ = 39 - 11 \sqrt{15} $$ **(d) Simplify $(3\sqrt{18} - 2\sqrt{27})(2\sqrt{32} + \sqrt{48})$:** First simplify surds: $$ \sqrt{18} = 3\sqrt{2}, \quad \sqrt{27} = 3\sqrt{3}, \quad \sqrt{32} = 4\sqrt{2}, \quad \sqrt{48} = 4\sqrt{3} $$ Rewrite expression: $$ (3 \times 3\sqrt{2} - 2 \times 3\sqrt{3})(2 \times 4\sqrt{2} + 4\sqrt{3}) = (9\sqrt{2} - 6\sqrt{3})(8\sqrt{2} + 4\sqrt{3}) $$ Multiply: $$ = 9\sqrt{2} \times 8\sqrt{2} + 9\sqrt{2} \times 4\sqrt{3} - 6\sqrt{3} \times 8\sqrt{2} - 6\sqrt{3} \times 4\sqrt{3} $$ $$ = 72 \times (\sqrt{2})^2 + 36 \sqrt{6} - 48 \sqrt{6} - 24 \times (\sqrt{3})^2 $$ $$ = 72 \times 2 + (36 - 48) \sqrt{6} - 24 \times 3 $$ $$ = 144 - 12 \sqrt{6} - 72 $$ $$ = (144 - 72) - 12 \sqrt{6} = 72 - 12 \sqrt{6} $$ 4. **Final Answers:** - (\sqrt{3} - 1)(\sqrt{3} + 1) = $2$ - (3 - \sqrt{2})(2 + 3\sqrt{2}) = $7\sqrt{2}$ - (3\sqrt{5} - \sqrt{3})(2\sqrt{5} - 3\sqrt{3}) = $39 - 11\sqrt{15}$ - (3\sqrt{18} - 2\sqrt{27})(2\sqrt{32} + \sqrt{48}) = $72 - 12\sqrt{6}$ These steps show how to multiply surds by applying distributive property and simplifying radicals carefully.