Which list contains all complex roots of z^4 = -16?

• A. √2 + i√2, -√2 + i√2, -√2 - i√2, √2 - i√2
• B. 2, -2, 2i, -2i
• C. 1 ± i, -1 ± i
• D. √3 ± i, -√3 ± i

Answer: (A)

Solving z^4 = -16 uses expressing the complex number on the right in polar form and applying De Moivre’s theorem. The number -16 has modulus 16 and argument π (mod 2π). If z = re^{iθ}, then r^4 = 16, so r = 2. The angles must satisfy 4θ ≡ π (mod 2π), giving θ = (π + 2πk)/4 for k = 0,1,2,3. This yields θ = π/4, 3π/4, 5π/4, 7π/4, and the roots are z = 2e^{iθ} = √2 + i√2, -√2 + i√2, -√2 - i√2, √2 - i√2. These four numbers are exactly the roots, and they are all of magnitude 2 with those four equal-spaced angles. The other lists don’t satisfy z^4 = -16 because they have the wrong magnitudes or angles, so they do not produce -16 when raised to the fourth power.