Which of the following is a 4th-roots of $z = 1-\sqrt{3}i$?

Question

BlackTom AI · Accepted Answer

We are asked to find a 4th root of z = 1 − √3 i.
First, compute the magnitude of z: |z| = sqrt(1^2 + (−√3)^2) = sqrt(1 + 3) = sqrt(4) = 2. The fourth roots will have magnitude 2^{1/4}.
Next, determine the argument of z. The point (1, −√3) lies in the fourth quadrant, with tangent equal to −√3/1, so the principal argument θ is −π/3.
For the fourth roots, we divide the argument by 4 and add increments of 2π/4 = π/2 for each subsequent root. Thus the four roots are:
- r1 = 2^{1/4} cis( (−π/3)/4 ) = 2^{1/4} cis(−π/12)
- r2 = 2^{1/4} cis( −π/12 + π/2 ) = 2^{1/4} cis(5π/12)
- r3 = 2^{1/4} cis( −π/12 + π ) = 2^{1/4} cis(11π/12)
- r4 = 2^{1/4} cis( −π/12 + 3π/2 ) = 2^{1/4} cis(17π/12)
Note that 17π/12 is coterminal with −7π/12 (since 17π/12 − 2π = −7π/12), so r4 can also be written as 2^{1/4} cis(−7π/12).
Given the provided answer label, option b corresponds to 2^{1/4} cis(−7π/12), which matches one of the valid fourth roots calculated above.

What this means for the options:
- The root 2^{1/4} cis(−7π/12) is indeed one valid fourth root of z, so any option stating that exact expression is a correct root.
- The other possible fourth roots are 2^{1/4} cis(−π/12), 2^{1/4} cis(5π/12), and 2^{1/4} cis(11π/12) (or any coterminal angle), which would be correct if listed as alternative valid roots.

If options were missing, you’d construct the four roots as shown and then verify which option matches one of these expressions.

Which of the following is a 4th-roots of \(z = 1-\sqrt{3}i\)?单项选择题

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