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x + 1/x = 2: value of x^m + 1/x^n for large powers

The same question, asked 2 times: 19 Jul 2023 S4*, 18 Sep 2024 S2. Every wording below, oldest first.

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  1. Question 1

    19 Jul 2023 S4

    67
    If (x+1x)=2(x+\frac{1}{x}) =2, then x7+1x117=x^7+\frac{1}{x^{117}}= _________.

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  2. Question 2

    18 Sep 2024 S2

    54
    If x+1x=2x + \frac{1}{x} = 2, then find the value of (x2723+1x3356)\left(x^{2723} + \frac{1}{x^{3356}}\right).

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