` a) ` ` \sqrt{50} - 3\sqrt{8} + \sqrt{(\sqrt{2} + 1)^2} `
` = 5\sqrt{2} - 6\sqrt{2} + \sqrt{2} + 1 `
` = (5 - 6 + 1)\sqrt{2} + 1 `
` = 1 `
` b) ` ` \sqrt{8} + \frac{7}{\sqrt{2} + 3} + \sqrt{(\sqrt{2} - 5)^2} `
` = 2\sqrt{2} + 3 - \sqrt{2} + 5 - \sqrt{2} `
` = (2 - 1 - 1)\sqrt{2} + 3 + 5 `
` = 8 `
` c) ` ` \sqrt{32} - 6\sqrt{2} + 2\sqrt{(1 - \sqrt{2})^2} `
` = 4\sqrt{2} - 6\sqrt{2} + 2(\sqrt{2} - 1) `
` = -2\sqrt{2} + 2\sqrt{2} - 2 `
` = -2 `
` d) ` ` (3\sqrt{5} - 1/2 \sqrt{80} + \sqrt{20}).(-\sqrt{5}) `
` = (3\sqrt5 - 2\sqrt{5} + 2\sqrt{5}).(-\sqrt{5}) `
` = 3\sqrt{5}.(-\sqrt{5}) `
` = 3.(-5) = -15 `
