Q: Which formula is used to find the $n$th 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 $n$th 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 $n$th 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 $n$th 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)

Study Guide - Practice Questions

Study Guide - Flashcards