which of the following expressions is a sixth root of unity? -1 i -i all of the above

Question

which of the following expressions is a sixth root of unity?
-1
i
-i
all of the above

in progress 0
2022-04-29T17:07:59+00:00 2 Answers 0

Answers ( 2 )

  1. Emery
    0
    2022-04-29T17:09:14+00:00

    Let sixth root of unity be x

    So [tex] x^{6}=1 [/tex]

    [tex] x^6-1=0 [/tex]

    [tex] (x^3)^2-1^2=0 [/tex]

    [tex] (x^3-1)(x^3+1)=0 [/tex][tex] x^3-1=0 $ or $ x^3+1=0 [/tex]

    [tex] (x-1)(x^2+x+1)=0 $ or $ (x+1)(x^2-x+1)=0 [/tex][tex] x-1=0 $ or $ x^2+x+1=0 $ or $ x+1=0 $ or $ x^2-x+1=0 [/tex]

    Using Quadratic formula for the quadratic equations we get

    [tex] x=1 $ or $ x=frac{-1pmsqrt{1-4} }{2} $ or $ x = -1 $ or $ x=frac{1pmsqrt{1-4} }{2} [/tex]

    [tex] x = 1 $ or $ x=frac{-1pmsqrt{3}i }{2} $ or $ x = -1 $ or $ x=frac{1pmsqrt{3}i }{2} [/tex]

    Only 1, out of the given option is the sixth root of unity.

    0
    2022-04-29T17:09:37+00:00

    -1 for apex i just answered this question and got it right

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