Q: If $z = 2 \left( \cos \theta + i \sin \theta \right)$, what is $z^3$ in exponential form?

$8 \left( \cos 3\theta + i \sin 3\theta \right)$

Q: Which of the following is the correct statement of De Moivre's Theorem?

$[r (\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta)$

Question 1

Given the complex number $z = 4 \left( \cos \left( \frac{\pi}{3} \right) + i \sin \left( \frac{\pi}{3} \right) \right)$, what is $z^2$ in polar form?

Accepted Answer

$16 \left( \cos \left( \frac{2\pi}{3} \right) + i \sin \left( \frac{2\pi}{3} \right) \right)$

Question 2

If $z = 2 \left( \cos 	heta + i \sin 	heta ight)$, what is $z^3$ in exponential form?

Accepted Answer

$8 \left( \cos 3	heta + i \sin 3	heta ight)$

Question 3

Which theorem allows you to compute powers of complex numbers in polar form efficiently?

Accepted Answer

De Moivre's Theorem

Question 4

Given $z = 3 \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$, find $z^4$ in rectangular form.

Accepted Answer

$81 \left( \cos \pi + i \sin \pi \right) = -81$

Question 5

If $z = r \left( \cos 	heta + i \sin 	heta ight)$, what is the modulus of $z^n$?

Accepted Answer

$r^n$

Question 6

A complex number $z = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)$. What is $z^3$ in rectangular form?

Accepted Answer

$8 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) = 8i$

Question 7

Which of the following is the correct statement of De Moivre's Theorem?

Accepted Answer

$[r (\cos 	heta + i \sin 	heta)]^n = r^n (\cos n	heta + i \sin n	heta)$

Question 8

If $z = 1 + i$, what is $z^4$?

Accepted Answer

$-4$

Question 9

A phasor in electrical engineering is represented as $z = 5 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)$. What is its value in rectangular form?

Accepted Answer

$5i$

Question 10

If $z = 2 \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$, what is $z^2$ in rectangular form?

Accepted Answer

$4 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) = 4i$

Question 11

Which of the following is the rectangular form of $z = 3 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$?

Accepted Answer

$-1.5 + 2.598i$

Question 12

If $z = r (\cos 	heta + i \sin 	heta)$, what is the argument of $z^n$?

Accepted Answer

$n	heta$

Question 13

A signal rotates in the complex plane with $z = 2 (\cos \omega t + i \sin \omega t)$. What is $z^5$?

Accepted Answer

$32 (\cos 5\omega t + i \sin 5\omega t)$

Question 14

If $z = 1 - i$, what is $z^2$?

Accepted Answer

$0 - 2$

Question 15

Given $z = 2 (\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})$, what is $z^6$?

Accepted Answer

$64$

Powers of Complex Numbers (De Moivre’s Theorem)

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