# 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
- DOnly when is an even number
- EWhen it is not an root of unity for some

**Q3: **

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

- A
- B1
- 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?

- A1
- B
- C
- D
- E

**Q11: **

Write a general form for the roots , giving your answer in polar form.

- A
- B
- C
- D
- E

**Q12: **

Find the quintic roots of unity.

- A1, , , ,
- B, , , ,
- C1, , , ,
- D1, , , ,
- E, , , ,

What is the value of their sum?

**Q13: **

Let be an th root of unity.

Which of the following is the correct relationship between and ?

- A
- B
- C
- D

Express in terms of positive powers of .

- A
- B
- C
- D
- E

**Q14: **

For how many pairs of real numbers does the relation hold?

**Q15: **

Find the cube roots of unity.

- A1, ,
- B1, ,
- C1, ,
- D1, ,
- E1, ,

Find the solutions to .

- A1, , , , ,
- B1, , , , ,
- C1, , , , ,
- D1, , , , ,
- E1, , , , ,

What is the relationship between the cubic roots of unity and the 6th roots of unity?

- A1 is the only common root between the cubic roots of unity and the 6th roots of unity.
- BAll of the cubic roots of unity are also 6th roots of unity.
- CAll of the cubic roots of unity and their complex conjugates are 6th roots of unity.
- DThe cubic roots of unity divided by 2 are 6th roots of unity.
- EThere are no common roots between the cubic roots of unity and the 6th roots of unity.

**Q16: **

Two regular polygons are inscribed in the same circle: the first has 1,731 sides and the second has 4,039. If the two polygons have at least one vertex in common, how many vertices in total will coincide?

**Q17: **

Two regular polygons are inscribed in the same circle where the first has 1,731 sides and the second has 4,039. If the two polygons have at least one vertex in common, how many vertices in total will coincide?

**Q18: **

Find the cube roots of unity in algebraic form.

- A, ,
- B, ,
- C1, ,
- D1, ,
- E, ,

Plot the roots on an Argand diagram.

- A
- B
- C
- D
- E

**Q19: **

What is the relationship between the cube roots of unity and the sixth roots of unity?

- ASixth roots of unity are double cube roots of unity.
- BThere is no relationship between sixth roots of unity and cube roots of unity.
- CAll sixth roots of unity are also cube roots of unity.
- DCube roots of unity are half sixth roots of unity.
- EAll cube roots of unity are also sixth roots of unity.

**Q20: **

Find the sum of the sixth roots of unity.