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x a perfect square of surds: find x^(1/n) + x^(-1/n)

This type of question, asked 4 times: CDS 1 2019, CDS 1 2022, CDS 2 2023, CDS 2 2026. All 4 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 1 2019

    92
    If a=7+43a = \sqrt{7 + 4\sqrt{3}}, then what is the value of a+1aa + \frac{1}{a} ?

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  2. Question 2

    CDS 1 2022

    3
    If x=7+43x = 7 + 4\sqrt{3}, then what is the value of x+1x\sqrt{x} + \frac{1}{\sqrt{x}} ?

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  3. Question 3

    CDS 2 2023

    12
    If x=97+563x = 97 + 56\sqrt{3}, then what is the value of x4+1x4\sqrt[4]{x} + \dfrac{1}{\sqrt[4]{x}} ?

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  4. Question 4

    CDS 2 2026

    10
    If x=11+230x = 11 + 2\sqrt{30}, then what is the value of x1/2+x−1/2x^{1/2} + x^{-1/2} ?

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