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Let z satisfy z⁴ = -1 + sqrt(3)*i. Which of the following statements is correct about its four roots?

1. z³ is purely real
2. z represents the vertices of a square of side 2^(1/4)
3. z⁹ is purely imaginary
4. z represents the vertices of a square of side 2^(3/4)

Correct answer: z represents the vertices of a square of side 2^(3/4)

Solution

The four roots lie on a circle of radius 2^(1/4); a square inscribed in a circle of radius R has side R*sqrt(2), giving side 2^(1/4)*2^(1/2) = 2^(3/4).

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