1. Simplify each expression step-by-step: i. Simplify \(2\sqrt{8} + \sqrt{18} - \frac{6}{\sqrt{2}}\): - \(\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\) - So, \(2\sqrt{8} = 2 \times 2\sqrt{2} = 4\sqrt{2}\) - \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\) - \(\frac{6}{\sqrt{2}} = \frac{6 \sqrt{2}}{2} = 3\sqrt{2}\) - Summing: \(4\sqrt{2} + 3\sqrt{2} - 3\sqrt{2} = 4\sqrt{2}\) ii. Simplify \(3\sqrt{8} \times 2\sqrt{32} - \frac{10}{\sqrt{2}}\): - \(\sqrt{8} = 2\sqrt{2}\), so \(3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2}\) - \(\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\), so \(2\sqrt{32} = 2 \times 4\sqrt{2} = 8\sqrt{2}\) - Multiply: \(6\sqrt{2} \times 8\sqrt{2} = 6 \times 8 \times (\sqrt{2} \times \sqrt{2}) = 48 \times 2 = 96\) - \(\frac{10}{\sqrt{2}} = \frac{10 \sqrt{2}}{2} = 5\sqrt{2}\) - Final: \(96 - 5\sqrt{2}\) iii. Simplify \(\sqrt{24.5} - \sqrt{12.5}\): - Express as decimal: \(\sqrt{24.5} \approx 4.95\), \(\sqrt{12.5} \approx 3.54\) - Subtract: \(4.95 - 3.54 = 1.41\) (approximate) 2. Simplify \( \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \) and express in form \(a + b\sqrt{3}\): - Multiply numerator and denominator by conjugate \(3 - \sqrt{3}\): \[ \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \times \frac{3 - \sqrt{3}}{3 - \sqrt{3}} = \frac{(3 - \sqrt{3})^2}{3^2 - (\sqrt{3})^2} = \frac{9 - 2 \times 3 \sqrt{3} + 3}{9 - 3} = \frac{12 - 6\sqrt{3}}{6} = 2 - \sqrt{3} \] - Therefore, \(a = 2\), \(b = -1\) 3. Simplify \( \frac{\sqrt{6} + \sqrt{3}}{\sqrt{2} + 1} \) writing answer in terms of \( \sqrt{3} \): - Multiply numerator and denominator by conjugate \(\sqrt{2} - 1\): \[ \frac{(\sqrt{6} + \sqrt{3})(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{\sqrt{6}\sqrt{2} - \sqrt{6} + \sqrt{3}\sqrt{2} - \sqrt{3}}{2 - 1} = \sqrt{12} - \sqrt{6} + \sqrt{6} - \sqrt{3} = \sqrt{12} - \sqrt{3} \] - Simplify \(\sqrt{12} = 2\sqrt{3}\), so expression becomes: \[ 2\sqrt{3} - \sqrt{3} = \sqrt{3} \] 4. Solve equations: i. \(\frac{1}{x} = 2x + 3\) - Multiply both sides by \(x\): \[ 1 = 2x^2 + 3x \] - Rearrange: \[ 2x^2 + 3x - 1 = 0 \] - Use quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a=2, b=3, c=-1\): \[ x = \frac{-3 \pm \sqrt{9 + 8}}{4} = \frac{-3 \pm \sqrt{17}}{4} \] ii. \(3x - \frac{5}{x} = 2\) - Multiply both sides by \(x\): \[ 3x^2 - 5 = 2x \] - Rearrange: \[ 3x^2 - 2x - 5 = 0 \] - Use quadratic formula with \(a=3, b=-2, c=-5\): \[ x = \frac{2 \pm \sqrt{4 + 60}}{6} = \frac{2 \pm \sqrt{64}}{6} = \frac{2 \pm 8}{6} \] - Solutions: \[ x = \frac{10}{6} = \frac{5}{3} \quad \text{or} \quad x = \frac{-6}{6} = -1 \] iii. Solve \(5y^2 + 7y - 6 = 0\): - Use quadratic formula with \(a=5, b=7, c=-6\): \[ y = \frac{-7 \pm \sqrt{49 + 120}}{10} = \frac{-7 \pm \sqrt{169}}{10} = \frac{-7 \pm 13}{10} \] - Solutions: \[ y = \frac{6}{10} = 0.6 \quad \text{or} \quad y = \frac{-20}{10} = -2 \] Final answers: i. \(4\sqrt{2}\) ii. \(96 - 5\sqrt{2}\) iii. Approximately \(1.41\) b. \(2 - \sqrt{3}\), where \(a=2\), \(b=-1\) c. \(\sqrt{3}\) d.i. \(x = \frac{-3 \pm \sqrt{17}}{4}\) d.ii. \(x = \frac{5}{3} \text{ or } -1\) d.iii. \(y = 0.6 \text{ or } -2\)