化簡∛(20-14√2)?

假設 a - b√2 = ∛(20 - 14√2)，則
(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，因為 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，因為 1² - 4(1)(1) < 0)
==> b = 1
所以 a = 2b = 2
所以 ∛(20 - 14√2) = 2 - √2

2-sqrt(2)