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?

Accepted Answer

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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