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Given x + 1/x (directly or from an equation): value of x^n + 1/x^n

This type of question, asked 6 times: 15 Nov 2020 Tier II, 29 Jan 2022 Tier II, 3 Feb 2022 Tier II, 8 Aug 2022 Tier II, 2 Mar 2023 Tier II, 18 Jan 2026 Tier II. All 6 questions below, oldest first.

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  1. Question 1

    15 Nov 2020 Tier II

    76
    Given that x8−34x4+1=0x^8 - 34x^4 + 1 = 0, x>0x > 0. What is the value of (x3+x−3)(x^3 + x^{-3})?

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  2. Question 2

    29 Jan 2022 Tier II

    92
    If x2−7x+1=0x^2 - \sqrt{7}x + 1 = 0, then what is the value of x5+1x5x^5 + \frac{1}{x^5}?

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  3. Question 3

    3 Feb 2022 Tier II

    78
    If x+1x=3x + \frac{1}{x} = 3, x≠0x \ne 0, then the value of x7+1x7x^7 + \frac{1}{x^7} is:

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  4. Question 4

    8 Aug 2022 Tier II

    34
    If (x+1x)2=3\left(x + \frac{1}{x}\right)^2 = 3, then what is the value of x6+x−6x^6 + x^{-6}?

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  5. Question 5

    2 Mar 2023 Tier II

    30
    If 1x+x=4\frac{1}{x} + x = 4, then find 1x2+x2\frac{1}{x^2} + x^2.

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  6. Question 6

    18 Jan 2026 Tier II

    14
    If x+1x=23x + \frac{1}{x} = 2\sqrt{3}, then find the value of x3+1x3x^3 + \frac{1}{x^3}.

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