# Worksheet: The nth Roots of Unity

In this worksheet, we will practice using de Moivre's theorem to find the nth roots of unity and exploring their properties.

Q1:

Which of the following is a general form for the roots in exponential form?

• A
• B
• C
• D
• E

Q2:

Let be an root of unity. When can we define as a primitive root of unity?

• AWhen it is an root of unity for some
• BWhen it is an root of unity, where is a prime number
• COnly when is a prime number
• DWhen it is not an root of unity for some
• EOnly when is an even number

Q3:

If is a primitive root of unity, which of the following expressions is equivalent to ?

• A
• B
• C
• D1
• E

Q4:

Which of the following is one of the roots of unity in Cartesian form?

• A
• B
• C
• D
• E

Q5:

Let be one of the quintic roots of unity. Which of the following is an equivalent expression to ?

• A
• B
• C
• D
• E

Q6:

Let be an root of unity and be a positive integer. Which of the following is not an equivalent expression for ?

• A
• B
• C
• D
• E

Q7:

Let be an root of unity, where is even. Which of the following expressions is equivalent to ?

• A
• B
• C
• D
• E

Q8:

How many of the 8th roots of unity are also 12th roots of unity?

Q9:

What is the general form for the 10th roots of unity in polar form?

• A
• B
• C
• D
• E

Using the general form for 10th roots of unity, identify the 10th root of unity for the case where .

• A
• B
• C
• D
• E

Q10:

Which of the following is not one of the cube roots of unity?

• A
• B
• C
• D
• E1