Using de Moivre’s theorem to compute we find that the answer is


None of the others

To solve using de Moivre’s theorem, we first need to convert the complex number into polar form. The modulus is computed as . The argument can be found using (in the third quadrant).

Now in polar form, the complex number is . According to de Moivre’s theorem, we can raise it to the power of 9:

Evaluating and simplifying the angle , which can be adjusted to (as ), we can find .

Combining these, we get:

So, the correct answer is .

The next step would be ensuring understanding of de Moivre’s theorem. It’s a nifty formula in mathematics that allows you to raise complex numbers to powers swiftly by converting them into polar coordinates. Remember, turning complex numbers into their polar form transforms multiplication and exponentiation into much simpler trigonometric calculations—just keep an eye on those angles!

When dealing with complex numbers, always pay attention to the quadrant your result lands in. Keep track of your angles, especially when converting back and forth between radians and degrees, as small mistakes in quadrant identification can lead to significant errors in your final answer.