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 for a complex number $z=r(\cos	heta+i\sin	heta)$ and a positive integer $n$, we have $z^n=r^n(\cos n	heta+i\sin n	heta)$.

In this case, we have $z=11e^{i(\frac{c}{c+2\pi}+\frac{\pi}{6})}$, which can be written as $z=11(\cos(\frac{c}{c+2\pi}+\frac{\pi}{6})+i\sin(\frac{c}{c+2\pi}+\frac{\pi}{6}))$. To raise $z$ to the power of 14, we use DeMoivre's Theorem and get:

$z^{14}=11^{14}(\cos(14(\frac{c}{c+2\pi}+\frac{\pi}{6}))+i\sin(14(\frac{c}{c+2\pi}+\frac{\pi}{6})))$

Simplifying the argument, we have:

$14(\frac{c}{c+2\pi}+\frac{\pi}{6})=\frac{7c+7\pi}{c+2\pi}+\frac{7\pi}{3}$

Using the fact that $c$ lies between $-\pi$ and $\pi$, we can determine that the answer is (B) $11e^{i(\frac{7c+21\pi}{3c+6\pi})}$.

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