The portal has been deactivated. Please contact your portal admin.

Lesson Worksheet: Cube Roots of Unity Mathematics

In this worksheet, we will practice identifying the cubic roots of unity using de Moivre’s theorem.

Q1:

Find all the values of 𝑧 for which 𝑧=1.

  • A1,𝜋3+𝑖𝜋3cossin, cossin2𝜋3+𝑖2𝜋3
  • B1,𝜋2+𝑖𝜋2cossin, cossin𝜋2+𝑖𝜋2
  • C1,𝜋3+𝑖𝜋3cossin, cossin𝜋3+𝑖𝜋3
  • D1,2𝜋3+𝑖2𝜋3cossin, cossin2𝜋3+𝑖2𝜋3
  • E1,𝜋3+𝑖𝜋3cossin, cossin2𝜋3+𝑖2𝜋3

Q2:

Let 𝑧=𝑒 and 𝑧=𝑒 be the complex cubic roots of unity.

Evaluate 𝑧. How does this compare with 𝑧?

  • A2𝑒, 𝑧=2𝑧
  • B𝑒, 𝑧=𝑧
  • C2𝑒, 𝑧>2𝑧
  • D𝑒, 𝑧<𝑧
  • E𝑒, 𝑧=𝑧

Evaluate 𝑧. How does this compare with 𝑧?

  • A2𝑒, 𝑧>𝑧
  • B𝑒, 𝑧<𝑧
  • C2𝑒, 𝑧=2𝑧
  • D𝑒, 𝑧=𝑧
  • E𝑒, 𝑧=𝑧

Q3:

Evaluate 𝜔𝜔 where 𝜔 is a complex cube root of unity.

Q4:

Write 𝜔 in its simplest form, where 𝜔 is a primitive cube root of unity.

  • A𝜔
  • B𝜔
  • C1

Q5:

Write 𝜔 in its simplest form, where 𝜔 is a primitive cube root of unity.

  • A𝜔
  • B𝜔
  • C1

Q6:

Evaluate 1+𝜔, where 𝜔 is a primitive cube root of unity.

  • A1
  • B1
  • C𝜔
  • D𝜔
  • E𝜔

Q7:

Evaluate 8𝜔+15+5𝜔+1𝜔 where 𝜔 is a complex cube root of unity.

Q8:

Evaluate 9𝜔+9𝜔+6+6𝜔+6𝜔, where 𝜔 is a nontrivial cubic root of unity.

  • A100𝜔
  • B64𝜔
  • C100𝜔
  • D64𝜔

Q9:

Given that 𝜔is one of the complex cubic roots of unity, find the complex conjugate of 1+𝜔.

  • A1𝜔
  • B1+𝜔
  • C𝜔
  • D1𝜔

Q10:

What is (8+𝜔), where 𝜔 is a primitive cube root of unity?

  • A49
  • B48
  • C0
  • D57

This lesson includes 93 additional questions and 586 additional question variations for subscribers.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.