Use DeMoivre's Theorem to find the indicated power of the following complex number.
B)32
C)64
D)-64

Question

Use DeMoivre's Theorem to find the indicated power of the following complex number. 
B)32 
C)64 
D)-64

Quizplus · Accepted Answer

DeMoivre's Theorem states that $(\operatorname{cis}	heta)^n = \operatorname{cis}(n	heta)$ for any integer $n$. In this case, our complex number can be written in polar form as $\operatorname{cis}(190^\circ)$. To find the fourth power of this number, we apply DeMoivre's Theorem with $n=4$: $(\operatorname{cis}(190^\circ))^4 = \operatorname{cis}(4\cdot 190^\circ) = \operatorname{cis}(760^\circ)$. Simplifying $760^\circ$ to an equivalent angle in the interval $[0^\circ,360^\circ)$, we have $760^\circ = 2\cdot 360^\circ + 40^\circ$, so $\operatorname{cis}(760^\circ) = \operatorname{cis}(40^\circ)$. Converting back to rectangular form, we find that $\operatorname{cis}(40^\circ) = \frac{1}{2}(1 + i\sqrt{3})$. Taking the absolute value of this complex number and squaring it gives us $\left(\frac{1}{2}ight)^2 + \left(\frac{\sqrt{3}}{2}ight)^2 = \frac{1}{4} + \frac{3}{4} = 1$, so the answer is $\boxed{	extbf{(C) }64}$.

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