Q: Which formula is used to raise a complex number in polar form to a power $n$?

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

Q: A complex number $z$ is given in polar form as $z = r (\cos \theta + i \sin \theta)$. Which of the following is true for $z^n$?

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

Q: Which of the following is NOT a correct application of De Moivre's Theorem?

$z^n = r^n (\cos n\theta - i \sin n\theta)$

Q: Which of the following is the correct expansion for $z^n$ using De Moivre's Theorem?

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

Question 1

Which formula is used to raise a complex number in polar form to a power $n$?

Accepted Answer

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

Question 2

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

Accepted Answer

$64 \left( \cos 2\pi + i \sin 2\pi \right)$

Question 3

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

Accepted Answer

$r^n$

Question 4

De Moivre's Theorem is most useful for which of the following operations?

Accepted Answer

Raising a complex number in polar form to an integer power

Question 5

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

Accepted Answer

$16 \left( \cos \pi + i \sin \pi \right)$

Question 6

What is the rectangular form of $8 \left( \cos \pi + i \sin \pi \right)$?

Accepted Answer

$-8$

Question 7

If $z = 5 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)$, what is $z^2$?

Accepted Answer

$25 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$

Question 8

A complex number $z$ is given in polar form as $z = r (\cos 	heta + i \sin 	heta)$. Which of the following is true for $z^n$?

Accepted Answer

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

Question 9

Which of the following is NOT a correct application of De Moivre's Theorem?

Accepted Answer

$z^n = r^n (\cos n	heta - i \sin n	heta)$

Question 10

A medical imaging device uses the complex number $z = 3 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)$. What is $z^2$?

Accepted Answer

$9 \left( \cos \pi + i \sin \pi \right)$

Question 11

If $z = 2 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$, what is the value of $z^6$?

Accepted Answer

$64 \left( \cos 2\pi + i \sin 2\pi \right)$

Question 12

A signal in electrical engineering is represented by $z = 5 \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$. What is $z^2$ in rectangular form?

Question 13

Which of the following is the correct expansion for $z^n$ using De Moivre's Theorem?

Accepted Answer

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

Powers of Complex Numbers (De Moivre’s Theorem)

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