Q: Which of the following correctly states DeMoivre’s Theorem for a complex number $$z = r(\cos \theta + i \sin \theta)$$ and a positive integer $$n$$?

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

Q: Which formula gives all $$n$$ distinct $$nth $$roots of a nonzero complex number $$r(\cos \theta + i \sin \theta)$$?

$$\sqrt[n]{r} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right)$$ for $$k = 0, 1, ..., n-1$$

Q: If $$z = 4 + 4i$$, what is the modulus $$r$$ and argument $$\theta of$$ $$z in $$polar form?

$$r = 4\sqrt{2}, \theta = \frac{\pi}{4}$$

Question 1

Which of the following correctly states DeMoivre’s Theorem for a complex number $$z = r(\cos 	heta + i \sin 	heta)$$ and a positive integer $$n$$?

Accepted Answer

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

Question 2

Given $$z = 2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})$$, what is $$z^3$$ using DeMoivre’s Theorem?

Accepted Answer

$$8(\cos \pi + i \sin \pi)$$

Question 3

Which formula gives all $$n$$ distinct $$nth $$roots of a nonzero complex number $$r(\cos 	heta + i \sin 	heta)$$?

Accepted Answer

$$\sqrt[n]{r} \left( \cos \left( \frac{	heta + 2k\pi}{n} ight) + i \sin \left( \frac{	heta + 2k\pi}{n} ight) ight)$$ for $$k = 0, 1, ..., n-1$$

Question 4

How many distinct fourth roots does the complex number $$16$$ have?

Accepted Answer

4

Question 5

Which of the following is a correct expression for the $$nth $$roots of unity?

Accepted Answer

$$\cos \left( \frac{2k\pi}{n} \right) + i \sin \left( \frac{2k\pi}{n} \right)$$ for $$k = 0, 1, ..., n-1$$

Question 6

If $$z = 4 + 4i$$, what is the modulus $$r$$ and argument $$	heta of$$ $$z in $$polar form?

Accepted Answer

$$r = 4\sqrt{2}, 	heta = \frac{\pi}{4}$$

Question 7

Which of the following statements is TRUE regarding the roots of complex numbers?

Accepted Answer

There is no way to designate one solution as the principal root; all $$nth $$roots are equally valid.

Question 8

Find all cube roots of $$8 in $$polar form.

Accepted Answer

$$2 \left( \cos \frac{2k\pi}{3} + i \sin \frac{2k\pi}{3} \right)$$ for $$k = 0, 1, 2$$

Question 9

A chemical reaction produces a complex number $$z = 5(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}). $$What is $$z^4$$?

Accepted Answer

$$625(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3})$$

Question 10

Which of the following best describes the geometric location of the $$nth $$roots of unity in the complex plane?

Accepted Answer

They are equally spaced points on the unit circle.

Question 11

Find the principal value of $$(-\sqrt{3} + i)^4$ using DeMoivre’s Theorem.

Accepted Answer

$$16(\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3})$$

Question 12

A medical imaging device uses the fifth roots of $$4 + 4i. $$Which formula gives all possible roots?

Accepted Answer

$$\sqrt[5]{8}(\cos \frac{\pi/4 + 2k\pi}{5} + i \sin \frac{\pi/4 + 2k\pi}{5})$$ for $$k = 0, 1, 2, 3, 4$$

Question 13

Which of the following is NOT a valid $$nth $$root of unity for $$n = 4$$?

Accepted Answer

$$\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}$$

Question 14

If $$z = r(\cos 	heta + i \sin 	heta)$$, which of the following is the correct polar form for $$z^2$$?

Accepted Answer

$$r^2$$(\cos 2	heta + i \sin 2	heta)$$

Question 15

A signal processing algorithm requires the cube roots of $$-8. $$Which of the following gives all cube roots in polar form?

Accepted Answer

$$2(\cos \frac{\pi + 2k\pi}{3} + i \sin \frac{\pi + 2k\pi}{3})$$ for $$k = 0, 1, 2$$

Question 16

Which of the following best explains why the terminology of 'positive' and 'negative' roots does not apply to complex numbers?

Accepted Answer

Because 'positive' and 'negative' are not meaningful terms in the complex plane.

DeMoivre’s Theorem and nth Roots of Complex Numbers

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