Question

化简∛(20-14√2)?

Answer 1

假设 a - b√2 = ∛(20 - 14√2)，则 (a - b√2)³ = 20 - 14√2 ==> a³ - 3a²b√2 + 6ab² - 2b³√2 = 20 - 14√2 所以 a³ + 6ab² = 20 ==> 7a³ + 42ab² = 140 ⋯⋯⋯⋯ ① 3a²b + 2b³ = 14 ==> 30a²b + 20b³ = 140⋯⋯⋯⋯② 即 7a³ + 42ab² = 30a²b + 20b³ ==> 7a³ - 30a²b + 42ab² - 20b³ = 0 ==> (a - 2b)(7a² - 16ab + 10b²) = 0 ==> a - 2b = 0 或 7a² - 16ab + 10b² = 0 (不合，因为 16² - 4x7x10 < 0) ==> a = 2b 代入 ① 得 (2b)³ + 6(2b)b² = 20 ==> 8b³ + 12b³ = 20 ==> 20b³ = 20 ==> b³ - 1 = 0 ==> (b - 1)(b² + b + 1) = 0 ==> b - 1 = 0 或 b² + b + 1 = 0 (不合，因为 1² - 4(1)(1) < 0) ==> b = 1 所以 a = 2b = 2 所以 ∛(20 - 14√2) = 2 - √2
2-sqrt(2)