# Lesson Worksheet: The 𝑛th Roots of Unity Mathematics

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

Q1:

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

• A
• B
• C
• D
• E

Q2:

Find the quintic roots of unity.

• A1, , , ,
• B, , , ,
• C1, , , ,
• D1, , , ,
• E, , , ,

What is the value of their sum?

Q3:

Let be an th root of unity.

Which of the following is the correct relationship between and ?

• A
• B
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• D

Express in terms of positive powers of .

• A
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• E

Q4:

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

• A
• B
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• D
• E

Q5:

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.

Q6:

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?

Q7:

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

• A
• B1
• C
• D
• E

Q8:

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

Q9:

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

Q10:

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

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• B
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• E

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

• A
• B
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• D
• E

This lesson includes 23 additional questions and 50 additional question variations for subscribers.