Final answer:
None of the given options, (a), (b), (c), or (d), correctly represent a cube root of 2i when considering their angles in relation to the permissible values derived from dividing the argument of 2i by 3 and taking the cube root of the modulus.
Step-by-step explanation:
The question asks which expression is a cube root of 2i. To find the cube roots of a complex number, one approach is to express the number in polar form and then apply De Moivre's Theorem. The polar form of a complex number is r(cos(θ)+isin(θ)), where r is the modulus and θ is the argument of the complex number. The complex number 2i has a modulus of 2 and an argument of 90 degrees (or π/2 radians) because it lies on the positive imaginary axis.
To find the cube roots, we divide the argument by 3 and take the cube root of the modulus. The modulus of 2i, which is 2, gives us a cube root of 3√2 when taken to the one-third power. We then divide the argument of 90 degrees by 3, which gives us 30 degrees. However, there are three cube roots of a complex number, equidistant from each other on the complex plane, so we need to add 120 degrees (or 2π/3 radians) for each subsequent root. Thus, the arguments of the cube roots will be 30, 150, and 270 degrees.
None of the provided options exactly match these arguments. However, since we can express angles equivalent to the principal angle by adding or subtracting full circles (360 degrees), we can find which of the given options has an angle that is equivalent to one of the cube roots' angles. The angle of 210 degrees in option (a) is equivalent to 150 degrees because 210 - 150 = 60, which is not a multiple of 360. So option (a) cannot be right. The same applies to option (b), (c), and (d), none of which is an equivalent angle to 30, 150, or 270 degrees.
Therefore, none of the provided options is correct for the cube root of 2i.