Use DeMoivre's Theorem to find the indicated power of the complex number.Write the result in standard form.

Question

Use DeMoivre's Theorem to find the indicated power of the complex number.Write the result in standard form. ​  ​

Quizplus · Accepted Answer

DeMoivre's Theorem states that $(\operatorname{cis} 	heta)^n = \operatorname{cis} (n	heta)$ for any integer $n$ and any angle $	heta$ in radians.

In this case, we have $11\operatorname{cis}\left(\frac{2\pi}{9}ight)$. To find $(11\operatorname{cis}\left(\frac{2\pi}{9}ight))^6$, we can use DeMoivre's Theorem with $n=6$:

$$(11\operatorname{cis}\left(\frac{2\pi}{9}ight))^6 = 11^6 \operatorname{cis} \left(6\cdot \frac{2\pi}{9}ight) = 11^6 \operatorname{cis} \left(\frac{4\pi}{3}ight)$$

We can simplify the angle $\frac{4\pi}{3}$ as follows:

$$\frac{4\pi}{3} = \pi + \frac{\pi}{3} = \operatorname{cis}(\pi) \cdot \operatorname{cis}\left(\frac{\pi}{3}ight) = -1 \cdot \frac{1}{2} + \frac{\sqrt{3}}{2}i $$$$\Rightarrow 11^6 \operatorname{cis} \left(\frac{4\pi}{3}ight) = 11^6 \left(-\frac{1}{2} + \frac{\sqrt{3}}{2}iight) = \boxed{	extbf{(D) }11e^{acce61f408dd19f49016014c15cb1}}$$

Use DeMoivre's Theorem to Find the Indicated Power of the Complex