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x + 1/x given, x > 1: find x - 1/x first, then x^n - 1/x^n

This type of question, asked 9 times: 18 Jul 2023 S3, 19 Jul 2023 S1, 20 Jul 2023 S4, 26 Jul 2023 S3, 27 Jul 2023 S3, 11 Sep 2024 S1, 17 Sep 2024 S3, 19 Sep 2024 S2, 22 Sep 2025 S1. All 9 questions below, oldest first.

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  1. Question 1

    18 Jul 2023 S3

    73
    If (x+1x)=5\left(x + \frac{1}{x}\right) = 5, and x>1x > 1, what is the value of (x8−1x8)\left(x^8 - \frac{1}{x^8}\right)?

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  2. Question 2

    19 Jul 2023 S1

    54
    If (x+1x)=22\left(x + \frac{1}{x}\right) = 2\sqrt{2}, and x>1x > 1, what is the value of (x6−1x6)\left(x^6 - \frac{1}{x^6}\right)?

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  3. Question 3

    20 Jul 2023 S4

    60
    If (x+1x)=32\left(x + \frac{1}{x}\right) = 3\sqrt{2}, and x>1x > 1, what is the value of (x8−1x8)\left(x^8 - \frac{1}{x^8}\right)?

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  4. Question 4

    26 Jul 2023 S3

    69
    If (x+1x)=6\left(x + \frac{1}{x}\right) = \sqrt{6}, and x>1x > 1, what is the value of (x8−1x8)\left(x^8 - \frac{1}{x^8}\right)?

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  5. Question 5

    27 Jul 2023 S3

    60
    If (x+1x)=6\left(x + \frac{1}{x}\right) = 6, and x>1x > 1, find the value of (x2−1x2)\left(x^2 - \frac{1}{x^2}\right).

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  6. Question 6

    11 Sep 2024 S1

    56
    If x+1x=210x + \frac{1}{x} = 2\sqrt{10}, where x>1x > 1, then the value of (x3−1x3)\left(x^3 - \frac{1}{x^3}\right) is:

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  7. Question 7

    17 Sep 2024 S3

    53
    If x is an integer such that x+1x=658x + \frac{1}{x} = \frac{65}{8}, then find the value of x−1xx - \frac{1}{x}.

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  8. Question 8

    19 Sep 2024 S2

    70
    If (x+1x)=7,x>0\left(x + \frac{1}{x}\right) = 7, x > 0. The positive value of (x−1x)\left(x - \frac{1}{x}\right) is:

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  9. Question 9

    22 Sep 2025 S1

    71
    Given, x+1x=5x + \frac{1}{x} = 5, then determine the value of 2xx2−1\frac{2x}{x^2 - 1}

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