1. Problem 14(a): Simplify $\sqrt{32} + \sqrt{98}$. - $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$. - $\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}$. - Therefore, $\sqrt{32} + \sqrt{98} = 4\sqrt{2} + 7\sqrt{2} = 11\sqrt{2}$. 2. Problem 14(b): Rationalize the denominator of $\frac{1}{\sqrt{2} + 1}$. - Multiply numerator and denominator by the conjugate: $\frac{1}{\sqrt{2}+1} \times \frac{\sqrt{2}-1}{\sqrt{2}-1} = \frac{\sqrt{2} - 1}{(\sqrt{2})^2 - 1^2} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1$. 3. Problem 15(a): Rationalize denominator and simplify $\frac{9}{2 + \sqrt{7}}$. - Multiply numerator and denominator by the conjugate: $\frac{9}{2 + \sqrt{7}} \times \frac{2 - \sqrt{7}}{2 - \sqrt{7}} = \frac{9(2 - \sqrt{7})}{2^2 - (\sqrt{7})^2} = \frac{9(2 - \sqrt{7})}{4 - 7} = \frac{9(2 - \sqrt{7})}{-3} = -3(2 - \sqrt{7}) = -6 + 3\sqrt{7}$. 4. Problem 15(b): Rationalize denominator and simplify $\frac{42}{\sqrt{6}}$. - Multiply numerator and denominator by $\sqrt{6}$: $\frac{42}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{42\sqrt{6}}{6} = 7\sqrt{6}$. 5. Problem 16(a): Simplify $\sqrt{75} - \sqrt{3}$. - $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$. - So, $\sqrt{75} - \sqrt{3} = 5\sqrt{3} - \sqrt{3} = 4\sqrt{3}$. 6. Problem 16(b): Rationalize the denominator and simplify $\frac{8}{1 - \sqrt{5}}$. - Multiply numerator and denominator by the conjugate: $\frac{8}{1 - \sqrt{5}} \times \frac{1 + \sqrt{5}}{1 + \sqrt{5}} = \frac{8(1 + \sqrt{5})}{1 - 5} = \frac{8(1 + \sqrt{5})}{-4} = -2(1 + \sqrt{5}) = -2 - 2\sqrt{5}$.