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Given that x + 1/x = 2, prove that x² + 1/x² = x⁴ + 1/x⁴ = x⁸ + 1/x⁸. The common value of all three expressions is:

1. 2
2. 0
3. 4
4. 1

Correct answer: 2

Solution

The equation x + 1/x = 2 has the unique real solution x = 1 (since the AM-GM minimum of x + 1/x for x > 0 is 2, attained at x = 1). With x = 1, every power xⁿ = 1, so xⁿ + 1/xⁿ = 2 for all n. Hence all three expressions equal 2.

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