If x4+1/x4 =119 then find x3-1/x3
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x⁴ + 1/x⁴ = 119
(x² + 1/x² )² = x⁴ + 1/x⁴ + 2 = 119 + 2 = 121
x² + 1/x² = √121 = 11
(x - 1/x)² = x² + 1/x² - 2 = 11 - 2 = 9
x - 1/x = +- √9 = + 3 or - 3
case 1: x - 1/x = 3
(x - 1/x)³ = x³ - 1/x³ - 3 x * 1/x (x - 1/x)
27 = x³ - 1/x³ - 3 * 3
x³ - 1/x³ = 36
case 2: x - 1/x = -3
x³ - 1/x³ = -27 - 9 = -36
(x² + 1/x² )² = x⁴ + 1/x⁴ + 2 = 119 + 2 = 121
x² + 1/x² = √121 = 11
(x - 1/x)² = x² + 1/x² - 2 = 11 - 2 = 9
x - 1/x = +- √9 = + 3 or - 3
case 1: x - 1/x = 3
(x - 1/x)³ = x³ - 1/x³ - 3 x * 1/x (x - 1/x)
27 = x³ - 1/x³ - 3 * 3
x³ - 1/x³ = 36
case 2: x - 1/x = -3
x³ - 1/x³ = -27 - 9 = -36
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