1a. Rationalize the following expressions: 1. $$\frac{2}{3-\sqrt{2}} + \frac{1}{3+\sqrt{2}}$$ 2. $$\frac{3\sqrt{5} + 2\sqrt{3}}{2\sqrt{5} - 3\sqrt{3}}$$ 3. $$\frac{\sqrt{6} - 2\sqrt{3}}{\sqrt{6} + \sqrt{3}}$$ 4. $$\frac{10}{4\sqrt{18} - 3\sqrt{48}}$$ 5. $$\frac{\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} - 2\sqrt{2}}$$ 1b. Evaluate without tables: 6. $$\frac{\sqrt{50} - \sqrt{98}}{\sqrt{32}}$$ 7. $$\frac{1}{\sqrt{3} - 2} - \frac{1}{\sqrt{3} + 2}$$ 8. $$\frac{10\sqrt{32}}{\sqrt{18} - \sqrt{2}}$$ --- ### Step-by-step solutions: **1.** $$\frac{2}{3-\sqrt{2}} + \frac{1}{3+\sqrt{2}}$$ - Rationalize each denominator by multiplying numerator and denominator by the conjugate: $$\frac{2}{3-\sqrt{2}} \times \frac{3+\sqrt{2}}{3+\sqrt{2}} = \frac{2(3+\sqrt{2})}{(3)^2 - (\sqrt{2})^2} = \frac{6 + 2\sqrt{2}}{9 - 2} = \frac{6 + 2\sqrt{2}}{7}$$ $$\frac{1}{3+\sqrt{2}} \times \frac{3-\sqrt{2}}{3-\sqrt{2}} = \frac{3 - \sqrt{2}}{9 - 2} = \frac{3 - \sqrt{2}}{7}$$ - Add the two results: $$\frac{6 + 2\sqrt{2}}{7} + \frac{3 - \sqrt{2}}{7} = \frac{6 + 2\sqrt{2} + 3 - \sqrt{2}}{7} = \frac{9 + \sqrt{2}}{7}$$ **Answer:** $$\frac{9 + \sqrt{2}}{7}$$ **2.** $$\frac{3\sqrt{5} + 2\sqrt{3}}{2\sqrt{5} - 3\sqrt{3}}$$ - Multiply numerator and denominator by the conjugate of the denominator: $$\frac{3\sqrt{5} + 2\sqrt{3}}{2\sqrt{5} - 3\sqrt{3}} \times \frac{2\sqrt{5} + 3\sqrt{3}}{2\sqrt{5} + 3\sqrt{3}}$$ - Denominator: $$ (2\sqrt{5})^2 - (3\sqrt{3})^2 = 4 \times 5 - 9 \times 3 = 20 - 27 = -7 $$ - Numerator: $$ (3\sqrt{5} + 2\sqrt{3})(2\sqrt{5} + 3\sqrt{3}) = 3\sqrt{5} \times 2\sqrt{5} + 3\sqrt{5} \times 3\sqrt{3} + 2\sqrt{3} \times 2\sqrt{5} + 2\sqrt{3} \times 3\sqrt{3} $$ Calculate each term: $$3 \times 2 \times 5 = 30$$ $$3 \times 3 \times \sqrt{15} = 9\sqrt{15}$$ $$2 \times 2 \times \sqrt{15} = 4\sqrt{15}$$ $$2 \times 3 \times 3 = 18$$ Sum numerator: $$30 + 9\sqrt{15} + 4\sqrt{15} + 18 = 48 + 13\sqrt{15}$$ - Final expression: $$\frac{48 + 13\sqrt{15}}{-7} = -\frac{48 + 13\sqrt{15}}{7}$$ **Answer:** $$-\frac{48 + 13\sqrt{15}}{7}$$ **3.** $$\frac{\sqrt{6} - 2\sqrt{3}}{\sqrt{6} + \sqrt{3}}$$ - Multiply numerator and denominator by conjugate of denominator: $$\frac{\sqrt{6} - 2\sqrt{3}}{\sqrt{6} + \sqrt{3}} \times \frac{\sqrt{6} - \sqrt{3}}{\sqrt{6} - \sqrt{3}}$$ - Denominator: $$ (\sqrt{6})^2 - (\sqrt{3})^2 = 6 - 3 = 3 $$ - Numerator: $$ (\sqrt{6} - 2\sqrt{3})(\sqrt{6} - \sqrt{3}) = \sqrt{6} \times \sqrt{6} - \sqrt{6} \times \sqrt{3} - 2\sqrt{3} \times \sqrt{6} + 2\sqrt{3} \times \sqrt{3} $$ Calculate each term: $$6 - \sqrt{18} - 2\sqrt{18} + 2 \times 3 = 6 - 3\sqrt{2} - 6\sqrt{2} + 6 = 12 - 9\sqrt{2}$$ - Final expression: $$\frac{12 - 9\sqrt{2}}{3} = 4 - 3\sqrt{2}$$ **Answer:** $$4 - 3\sqrt{2}$$ **4.** $$\frac{10}{4\sqrt{18} - 3\sqrt{48}}$$ - Simplify radicals: $$\sqrt{18} = 3\sqrt{2}, \quad \sqrt{48} = 4\sqrt{3}$$ - Substitute: $$4 \times 3\sqrt{2} - 3 \times 4\sqrt{3} = 12\sqrt{2} - 12\sqrt{3}$$ - Factor denominator: $$12(\sqrt{2} - \sqrt{3})$$ - Rationalize denominator by multiplying numerator and denominator by conjugate: $$\frac{10}{12(\sqrt{2} - \sqrt{3})} \times \frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3}} = \frac{10(\sqrt{2} + \sqrt{3})}{12((\sqrt{2})^2 - (\sqrt{3})^2)} = \frac{10(\sqrt{2} + \sqrt{3})}{12(2 - 3)} = \frac{10(\sqrt{2} + \sqrt{3})}{12(-1)} = -\frac{10(\sqrt{2} + \sqrt{3})}{12}$$ - Simplify fraction: $$-\frac{5(\sqrt{2} + \sqrt{3})}{6}$$ **Answer:** $$-\frac{5(\sqrt{2} + \sqrt{3})}{6}$$ **5.** $$\frac{\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} - 2\sqrt{2}}$$ - Factor denominator: $$2(\sqrt{3} - \sqrt{2})$$ - Rewrite numerator: $$\sqrt{3} - 3\sqrt{2} = (\sqrt{3} - \sqrt{2}) - 2\sqrt{2}$$ - Split fraction: $$\frac{(\sqrt{3} - \sqrt{2}) - 2\sqrt{2}}{2(\sqrt{3} - \sqrt{2})} = \frac{\sqrt{3} - \sqrt{2}}{2(\sqrt{3} - \sqrt{2})} - \frac{2\sqrt{2}}{2(\sqrt{3} - \sqrt{2})} = \frac{1}{2} - \frac{\sqrt{2}}{\sqrt{3} - \sqrt{2}}$$ - Rationalize second term: $$\frac{\sqrt{2}}{\sqrt{3} - \sqrt{2}} \times \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} = \frac{\sqrt{2}(\sqrt{3} + \sqrt{2})}{3 - 2} = \sqrt{2}(\sqrt{3} + \sqrt{2}) = \sqrt{6} + 2$$ - Final expression: $$\frac{1}{2} - (\sqrt{6} + 2) = \frac{1}{2} - \sqrt{6} - 2 = -\sqrt{6} - \frac{3}{2}$$ **Answer:** $$-\sqrt{6} - \frac{3}{2}$$ **6.** $$\frac{\sqrt{50} - \sqrt{98}}{\sqrt{32}}$$ - Simplify radicals: $$\sqrt{50} = 5\sqrt{2}, \quad \sqrt{98} = 7\sqrt{2}, \quad \sqrt{32} = 4\sqrt{2}$$ - Substitute: $$\frac{5\sqrt{2} - 7\sqrt{2}}{4\sqrt{2}} = \frac{-2\sqrt{2}}{4\sqrt{2}} = \frac{-2}{4} = -\frac{1}{2}$$ **Answer:** $$-\frac{1}{2}$$ **7.** $$\frac{1}{\sqrt{3} - 2} - \frac{1}{\sqrt{3} + 2}$$ - Find common denominator: $$(\sqrt{3} - 2)(\sqrt{3} + 2) = (\sqrt{3})^2 - 2^2 = 3 - 4 = -1$$ - Rewrite expression: $$\frac{1(\sqrt{3} + 2) - 1(\sqrt{3} - 2)}{-1} = \frac{\sqrt{3} + 2 - \sqrt{3} + 2}{-1} = \frac{4}{-1} = -4$$ **Answer:** $$-4$$ **8.** $$\frac{10\sqrt{32}}{\sqrt{18} - \sqrt{2}}$$ - Simplify radicals: $$\sqrt{32} = 4\sqrt{2}, \quad \sqrt{18} = 3\sqrt{2}$$ - Substitute: $$\frac{10 \times 4\sqrt{2}}{3\sqrt{2} - \sqrt{2}} = \frac{40\sqrt{2}}{2\sqrt{2}} = \frac{40\sqrt{2}}{2\sqrt{2}} = 20$$ **Answer:** $$20$$