The four roots of z^4 = -16 have arguments (angles) of which values?

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Multiple Choice

The four roots of z^4 = -16 have arguments (angles) of which values?

The idea is to use polar form and De Moivre’s theorem. Write -16 as 16cis(π) (magnitude 16, argument π). The fourth roots have magnitude 16^(1/4) = 2 and their arguments are (π + 2πk)/4 for k = 0, 1, 2, 3. This gives π/4, 3π/4, 5π/4, and 7π/4. These four angles are evenly spaced by π/2 around the circle and lie in the standard range [0, 2π). Angles like -π/4 or -3π/4 are coterminal with 7π/4 and 5π/4, respectively, but the conventional list uses the positive representatives. Therefore the correct set of arguments is π/4, 3π/4, 5π/4, 7π/4.

The idea is to use polar form and De Moivre’s theorem. Write -16 as 16cis(π) (magnitude 16, argument π). The fourth roots have magnitude 16^(1/4) = 2 and their arguments are (π + 2πk)/4 for k = 0, 1, 2, 3. This gives π/4, 3π/4, 5π/4, and 7π/4. These four angles are evenly spaced by π/2 around the circle and lie in the standard range [0, 2π). Angles like -π/4 or -3π/4 are coterminal with 7π/4 and 5π/4, respectively, but the conventional list uses the positive representatives. Therefore the correct set of arguments is π/4, 3π/4, 5π/4, 7π/4.

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