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High powers of the complex roots of a quadratic with surd coefficients (polar form)

This type of question, asked 3 times: 2021 Paper 11, 2021 Paper 16, 12 Apr 2023. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 11

    46
    If α\alpha and β\beta are the distinct roots of the equation x2+(3)1/4x+31/2=0x^{2}+ (3)^{1/4}x+3^{1/2}= 0, then the value of α96(α12−1)+β96(β12−1)\alpha^{96}\left( \alpha^{12}- 1 \right)+\beta^{96}\left( \beta^{12}- 1 \right) is equal to:

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  2. Question 2

    2021 Paper 16

    45
    If α\alpha and β\beta are the roots of equation x2+2014x+512=0x^{2}+20^{\frac{1}{4}}x +5^{\frac{1}{2}}= 0, then find (α8+β8)\left( \alpha^{8}+\beta^{8} \right)

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  3. Question 3

    12 Apr 2023

    62
    Let α,β\alpha,\beta be the roots of the quadratic equation x2+6x+3=0x^{2}+\sqrt{6}x + 3 = 0. Then α23+β23+α14+β14α15+β15+α10+β10\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}} is equal to

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