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Problem 9 - Entrance Test

If (x² + 1/x²) = 7, find x⁶ + 1/x⁶.

Correct: A

Let a = x + 1/x. Then x² + 1/x² = a² - 2 = 7 → a² = 9 → a = ±3. Now x³ + 1/x³ = a³ - 3a = 27 - 9 = 18 (taking a=3). Then x⁶ + 1/x⁶ = (x³)² + (1/x³)² = (x³ + 1/x³)² - 2 = 18² - 2 = 324 - 2 = 322.