7. Which of the following is not a consequence of de Moivre's theorem? a. \( \arg \left(z^{\prime \prime}\right)=n \arg (z) \) b. \( \left|z^{n}\right| \leq|z|^{n} \) c. \( z^{n}=r^{n}(\cos n \theta+i \sin n \theta) \) d. \( z^{n}=r^{\prime \prime}(\cos n \theta-i \sin n \theta) \) e. \( z^{-1}=r^{\prime}(\cos n \theta+i \sin n \theta) \)

De Moivre's theorem is a powerful tool in complex number theory, stating that for any complex number in polar form \( z = r(\cos \theta + i\sin \theta) \), raising \( z \) to the \( n \)-th power results in \( z^n = r^n (\cos(n\theta) + i\sin(n\theta)) \). This theorem beautifully connects algebra and trigonometry, allowing for easier calculations in fields like electrical engineering and wave analysis where complex numbers abound. When working with de Moivre's theorem, a common mistake is misinterpreting the argument and modulus. For instance, some might think \( |z^n| \) is simply \( |z| \cdot n \) instead of realizing it's actually \( |z|^n \). This misunderstanding can lead to inaccuracies when analyzing complex functions or signal processing problems. Always remember: raising a complex number affects the modulus multiplicatively, not linearly!