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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