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Reduce high powers of omega (cube root of unity) using omega^3 = 1

This type of question, asked 6 times: 3 May 2023 S2, 11 Apr 2026 S2, 15 Apr 2026 S2, 17 Apr 2026 S1, 18 Apr 2026 S1, 20 Apr 2026 S1. All 6 questions below, oldest first.

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  1. Question 1

    3 May 2023 S2

    149
    If z2+z+1=0z^{2}+z+1=0 then (z3+1z3)2+(z4+1z4)2=\left(z^{3}+\frac{1}{z^{3}}\right)^{2}+\left(z^{4}+\frac{1}{z^{4}}\right)^{2}= where z=ω=z=\omega= complex cube root of unity

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  2. Question 2

    11 Apr 2026 S2

    51
    If α\alpha and β\beta are the roots of the equation x2+x+1=0x^2 + x + 1 = 0 then α2026+β2026=\alpha^{2026} + \beta^{2026} =

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  3. Question 3

    15 Apr 2026 S2

    52
    If ω\omega is a complex cube root of unity, then the value of sin⁡[π(ω10+ω23)−π4]=\sin\left[\pi(\omega^{10} + \omega^{23}) - \dfrac{\pi}{4}\right] =

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  4. Question 4

    17 Apr 2026 S1

    51
    If α\alpha and β\beta are the distinct roots of the equation x2−x+1=0x^2 - x + 1 = 0, then the value of α200+β206+2\alpha^{200} + \beta^{206} + 2 is equal to

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  5. Question 5

    18 Apr 2026 S1

    51
    The value of (−1+i32)18+(−1−i32)18\left(\dfrac{-1 + i\sqrt{3}}{2}\right)^{18} + \left(\dfrac{-1 - i\sqrt{3}}{2}\right)^{18} is

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  6. Question 6

    20 Apr 2026 S1

    51
    If ww is a complex cube root of unity , then the value of w10−w7+w5−w2+1w^{10} - w^7 + w^5 - w^2 + 1 is.

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