Q: Which formula is used to find the $$nth $$roots of a complex number $$z = r \operatorname{cis} \theta$$?

$$\sqrt[n]{r} \operatorname{cis} \left( \frac{\theta + 360^\circ k}{n} \right)$$, where $$k = 0, 1, ..., n-1$$

Question 1

Given the complex number $$z = 2 \operatorname{cis} 45^\circ$$, what is one of the cube roots of $$z$$? (Recall $$\operatorname{cis} 	heta = \cos 	heta + i \sin 	heta)$$

Accepted Answer

$$\sqrt[3]{2} \operatorname{cis} 15^\circ$$

Question 2

Which formula is used to find the $$nth $$roots of a complex number $$z = r \operatorname{cis} 	heta$$?

Accepted Answer

$$\sqrt[n]{r} \operatorname{cis} \left( \frac{	heta + 360^\circ k}{n} ight)$$, where $$k = 0, 1, ..., n-1$$

Question 3

If $$z = 8 \operatorname{cis} 120^\circ$$, what is $$z^{1/3} in $$polar form?

Accepted Answer

$$2 \operatorname{cis} 40^\circ$$

Question 4

A complex number $$z = 27 \operatorname{cis} 60^\circ is $$given. What is one of its cube roots?

Accepted Answer

$$3 \operatorname{cis} 20^\circ$$

Question 5

Which of the following is the correct process to find all $$nth $$roots of a complex number $$z = r \operatorname{cis} 	heta$$?

Accepted Answer

Calculate $$\sqrt[n]{r}$$ and $$\operatorname{cis} \left( \frac{	heta + 360^\circ k}{n} ight)$$ for $$k = 0 to$$ $$n-1$$

Question 6

If $$z = 16 \operatorname{cis} 180^\circ$$, what is one of the fourth roots of $$z$$?

Accepted Answer

$$2 \operatorname{cis} 45^\circ$$

Question 7

A student finds $$z = 81 \operatorname{cis} 270^\circ. $$What is $$z^{1/4}$$?

Accepted Answer

$$3 \operatorname{cis} 67.5^\circ$$

Question 8

Which of the following is NOT a valid root of $$z = 1 \operatorname{cis} 0^\circ$$ when finding the cube roots?

Accepted Answer

$$1 \operatorname{cis} 180^\circ$$

Question 9

If $$z = 32 \operatorname{cis} 150^\circ$$, what is one of the fifth roots of $$z$$?

Accepted Answer

$$2 \operatorname{cis} 30^\circ$$

Question 10

A real-world application: An engineer needs to distribute three identical signals, each with a phase difference, using the cube roots of unity. What are the arguments (angles) of these signals in degrees?

Accepted Answer

$$0^\circ, 120^\circ, 240^\circ$$

Question 11

Given $$z = 4 \operatorname{cis} 90^\circ$$, what is $$z^{1/2}$$?

Accepted Answer

$$2 \operatorname{cis} 45^\circ$$

Question 12

Which of the following best describes the geometric interpretation of the $$nth $$roots of a complex number?

Accepted Answer

They are equally spaced points on a circle in the complex plane.

Question 13

If $$z = 1 \operatorname{cis} 180^\circ$$, what are the square roots of $$z$$?

Accepted Answer

$$1 \operatorname{cis} 90^\circ$$ and $$1 \operatorname{cis} 270^\circ$$

Question 14

A chemist uses the fourth roots of unity to model molecular symmetry. What are the arguments (angles) in degrees for these roots?

Accepted Answer

$$0^\circ, 90^\circ, 180^\circ, 270^\circ$$

Question 15

If $$z = 9 \operatorname{cis} 270^\circ$$, what is one of the square roots of $$z$$?

Accepted Answer

$$3 \operatorname{cis} 135^\circ$$

Powers and Roots of Complex Numbers (Trigonometric Form)

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