Which of the following options lists all of the solutions to the following equation?
𝑧
3
=
−
1
+
3
𝑖
.

Question

BlackTom AI · Accepted Answer

Question restatement: The problem asks for all solutions to z^3 = -1 + 3i. The provided answer list appears to include three candidate roots, but there is no separate list of answer choices labeled A, B, C, etc. We will evaluate the given options as potential cube roots.

Step-by-step evaluation:
- First, convert the target complex number to polar form. The complex number -1 + 3i has magnitude r = sqrt((-1)^2 + 3^2) = sqrt(10), and an argument theta = arctan(3/(-1)) relative to the positive real axis places it in the second quadrant. A precise principal value is theta = π - arctan(3) ≈ 3.1416 - 1.2490 ≈ 1.8926 radians (about 108.43 degrees).
- The cube roots of -1 + 3i will have modulus R = r^{1/3} = (sqrt(10))^{1/3} = 10^{1/6} ≈ 1.467, and their arguments will be (theta + 2πk)/3 for k = 0, 1, 2. Therefore the three roots should have magnitudes approximately 1.467 and arguments approximately:
  - k = 0: (1.8926)/3 ≈ 0.6309 rad ≈ 36.15°
  - k = 1: (1.8926 + 2π)/3 ≈ (1.8926 + 6.2832)/3 ≈ 2.7252 rad ≈ 156.23°
  - k = 2: (1.8926 + 4π)/3 ≈ (1.8926 + 12.5664)/3 ≈ 4.8197 rad ≈ 276.20°
- Now evaluate each given option:
  - Option 1: z ≈ (2/3) e^{i 2π/9}. The modulus is 2/3 ≈ 0.667, far from the expected 1.467. Its argument is 2π/9 ≈ 40°. Even if the argument were close, the modulus mismatch already rules it out. This option does not satisfy z^3 = -1 + 3i.
  - Option 2: z ≈ (2/3) e^{i 8π/9}. Modulus 0.667 again, with argument 8π/9 ≈ 160°. The modulus is incorrect, so this cannot be a cube root of -1 + 3i.
  - Option 3: z ≈ (2/3) e^{i 14π/9}. Modulus 0.667, argument 14π/9 ≈ 280°. The modulus is still wrong, so this is also not a valid cube root.
- Conceptual check: For any cube root z, we must have |z|^3 = |-1 + 3i| = sqrt(10), so |z| = sqrt[3](sqrt(10)) ≈ 1.467. None of the options have this modulus, indicating a fundamental mismatch.

Common misconceptions or errors illustrated by these options:
- Assuming the modulus of a cube root can be 2/3 for this number, which contradicts |z|^3 = sqrt(10).
- Selecting an argument that seems plausible numerically without matching the required modulus, leading to inconsistent polar coordinates.
- Treating the equation z^3 = w as if any small-angle exponentials with small moduli could satisfy the cube root condition, ignoring the essential modulus constraint.

In sum, each provided option fails the necessary modulus condition |z| = 10^{1/6} ≈ 1.467, and thus none are correct cube roots of -1 + 3i. The correct roots would have magnitude about 1.467 and arguments approximately 0.6309, 2.7252, and 4.8197 radians, which correspond to the three cube roots of -1 + 3i. The given choices do not match these values.

Which of the following options lists all of the solutions to the following equation? 𝑧 3 = − 1 + 3 𝑖 .单项选择题

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