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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.


Write a general form for the roots of 𝑧=1, giving your answer in polar form.

  • Acossin2𝜋𝑘𝑛+𝑖2𝜋𝑘𝑛
  • Bsincos2𝜋𝑘𝑛+𝑖2𝜋𝑘𝑛
  • Ccossin(2𝜋𝑘𝑛)+𝑖(2𝜋𝑘𝑛)
  • Dcossin2𝜋𝑛𝑘+𝑖2𝜋𝑛𝑘
  • E((2𝜋𝑘)+𝑖(2𝜋𝑘))cossin


Find the quintic roots of unity.

  • A1, 𝑒, 𝑒, 𝑒, 𝑒
  • B1, 𝑒, 𝑒, 𝑒, 𝑒
  • C1, 𝑒, 𝑒, 𝑒, 𝑒
  • D1, 𝑒, 𝑒, 𝑒, 𝑒
  • E1, 𝑒, 𝑒, 𝑒, 𝑒

What is the value of their sum?


Let 𝜔 be an 𝑛th root of unity.

Which of the following is the correct relationship between 𝜔 and 𝜔?

  • A𝜔=𝜔
  • B𝜔=(𝜔)
  • C𝜔=𝜔
  • D𝜔=(𝜔)

Express 𝜔 in terms of positive powers of 𝜔.

  • A𝜔
  • B𝜔
  • C𝜔
  • D𝜔
  • E𝜔


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

  • A𝜔
  • B1𝜔
  • C𝜔
  • D𝜔
  • E𝜔


Find the cube roots of unity.

  • A1, cossin(𝜋)+𝑖(𝜋), cossin(2𝜋)+𝑖(2𝜋)
  • B1, cossin𝜋3+𝑖𝜋3, cossin𝜋3+𝑖𝜋3
  • C1, sincos2𝜋3+𝑖2𝜋3, sincos2𝜋3+𝑖2𝜋3
  • D1, cossin2𝜋3𝑖2𝜋3, cossin2𝜋3𝑖2𝜋3
  • E1, cossin2𝜋3+𝑖2𝜋3, cossin2𝜋3+𝑖2𝜋3

Find the solutions to 𝑧=1.

  • A1, cossin2𝜋6+𝑖2𝜋6, cossin2𝜋3+𝑖2𝜋3, 1, cossin2𝜋3+𝑖2𝜋3, cossin2𝜋6+𝑖2𝜋6
  • B1, cossin(𝜋)𝑖(𝜋), cossin(2𝜋)𝑖(2𝜋), 1, cossin(4𝜋)+𝑖(4𝜋), cossin(5𝜋)+𝑖(5𝜋)
  • C1, cossin𝜋6𝑖𝜋6, cossin𝜋3𝑖𝜋3, 1, cossin𝜋3+𝑖𝜋3, cossin𝜋6+𝑖𝜋6
  • D1, cossin𝜋6+𝑖𝜋6, cossin𝜋3+𝑖𝜋3, 1, cossin𝜋3+𝑖𝜋3, cossin𝜋6+𝑖𝜋6
  • E1, cossin2𝜋6𝑖2𝜋6, cossin2𝜋3𝑖2𝜋3, 1, cossin2𝜋3+𝑖2𝜋3, cossin2𝜋6+𝑖2𝜋6

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.


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?


If 𝜔 is a primitive 6th root of unity, which of the following expressions is equivalent to 𝜔+𝜔+𝜔?

  • A1𝜔𝜔
  • B1
  • C𝜔+𝜔+𝜔
  • D12𝜔+𝜔+𝜔
  • E1+𝜔+𝜔


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


For how many pairs of real numbers (𝑎,𝑏) does the relation (𝑎+𝑏𝑖)=𝑎𝑏𝑖 hold?


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

  • A((2𝜋𝑘)+𝑖(2𝜋𝑘))cossin
  • Bcossin10𝜋𝑘2+𝑖10𝜋𝑘2
  • Ccossin(10𝜋𝑘)+𝑖(10𝜋𝑘)
  • Dsincos2𝜋𝑘10+𝑖2𝜋𝑘10
  • Ecossin2𝜋𝑘10+𝑖2𝜋𝑘10

Using the general form for 10th roots of unity, identify the 10th root of unity for the case where 𝑘=3.

  • Acossin(30𝜋)+𝑖(30𝜋)
  • Bcossin3𝜋5+𝑖3𝜋5
  • Ccossin3𝜋10+𝑖3𝜋10
  • Dcossin3𝜋5+𝑖3𝜋5
  • Ecossin3𝜋5𝑖3𝜋5

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

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