What does DeMoivre's Theorem describe?

DeMoivre's Theorem provides a powerful tool for computing powers and roots of complex numbers in polar form. It states that for a complex number expressed in polar coordinates as ( z = r(\cos(\theta) + i\sin(\theta)) ), raising ( z ) to the nth power can be computed using the formula ( z^n = r^n(\cos(n\theta) + i\sin(n\theta)) ).

The correct choice follows this form precisely, indicating that the modulus ( r ) is raised to the power of ( n ) and the angle ( \theta ) is multiplied by ( n ). This allows for efficient computation by transforming complex multiplications into simpler operations involving their magnitudes and angles.

In contrast, the other choices either misrepresent the relationship of angles and magnitudes in DeMoivre's Theorem or include inaccuracies in how the angles are handled or expressed. This incorrect framing would not yield the proper result expected from the theorem, emphasizing the importance of using the correct format stated in the theorem itself. Thus, choice B accurately encapsulates the theorem's application and guarantees valid calculations for the powers of complex numbers.