he principal argument of  (1−i√3)4(1−𝑖3)4 is (where  i=√−1𝑖=−1) Only one correct answerA.π3π3B.π6π6C.ZeroD.2π32π3PreviousClearMark for Review & NextNextInstructionsEnglishNot VisitedNot AnsweredAnsweredMarked for ReviewAnswered & Marked for review(will be considered for evaluation)MATHEMATICSSection-ISection-II6162636465666768697071727374757677787980

The principal argument of a complex number is the angle it makes with the positive real axis.

First, let's simplify (1−i√3)⁴.

The complex number 1−i√3 can be written in polar form as r(cos θ + i sin θ), where r is the magnitude of the complex number and θ is the argument of the complex number.

The magnitude r is √((1)² + (√3)²) = 2.

The argument θ is arctan(√3/1) = π/3.

So, 1−i√3 = 2(cos π/3 - i sin π/3).

By De Moivre's theorem, (1−i√3)⁴ = [2(cos π/3 - i sin π/3)]⁴ = 2⁴[cos 4π/3 - i sin 4π/3] = 16(cos 4π/3 - i sin 4π/3).

The principal argument of 16(cos 4π/3 - i sin 4π/3) is 4π/3.

However, the principal argument is usually taken to be in the interval (-π, π].

So, the principal argument of (1−i√3)⁴ is 4π/3 - 2π = 2π/3.

Therefore, the correct answer is D. 2π/3.