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 𝑧 = 1 in exponential form?

  • A 𝑒
  • B 𝑛 𝑒
  • C 𝑒
  • D 𝑒
  • E 𝑒

Q2:

Let 𝜔 be an 𝑛 t h root of unity. When can we define 𝜔 as a primitive 𝑛 t h root of unity?

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

Q3:

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

  • A 1 + 𝜔 + 𝜔
  • B 𝜔 + 𝜔 + 𝜔
  • C 1 2 𝜔 + 𝜔 + 𝜔
  • D1
  • E 1 𝜔 𝜔

Q4:

Which of the following is one of the 8 t h roots of unity in Cartesian form?

  • A 2 2 + 2 2 𝑖
  • B 2 2 𝑖
  • C 2 2 2 2 𝑖
  • D 3 2 1 2 𝑖
  • E 1 2 3 2 𝑖

Q5:

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

  • A 1 𝜔
  • B 𝜔
  • C 𝜔
  • D 𝜔
  • E 𝜔

Q6:

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

  • A 𝜔
  • B 1 𝜔
  • C 𝜔
  • D 𝜔
  • E 𝜔

Q7:

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

  • A 𝜔
  • B ( 𝜔 )
  • C 𝜔
  • D 𝜔
  • E 𝜔

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 s i n c o s 2 𝜋 𝑘 1 0 + 𝑖 2 𝜋 𝑘 1 0
  • B ( ( 2 𝜋 𝑘 ) + 𝑖 ( 2 𝜋 𝑘 ) ) c o s s i n
  • C c o s s i n 2 𝜋 𝑘 1 0 + 𝑖 2 𝜋 𝑘 1 0
  • D c o s s i n ( 1 0 𝜋 𝑘 ) + 𝑖 ( 1 0 𝜋 𝑘 )
  • E c o s s i n 1 0 𝜋 𝑘 2 + 𝑖 1 0 𝜋 𝑘 2

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

  • A c o s s i n ( 3 0 𝜋 ) + 𝑖 ( 3 0 𝜋 )
  • B c o s s i n 3 𝜋 5 + 𝑖 3 𝜋 5
  • C c o s s i n 3 𝜋 1 0 + 𝑖 3 𝜋 1 0
  • D c o s s i n 3 𝜋 5 𝑖 3 𝜋 5
  • E c o s s i n 3 𝜋 5 + 𝑖 3 𝜋 5

Q10:

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

  • A c o s s i n 2 𝜋 3 + 𝑖 2 𝜋 3
  • B c o s s i n 2 𝜋 3 𝑖 2 𝜋 3
  • C c o s s i n 𝜋 3 + 𝑖 𝜋 3
  • D c o s s i n 2 𝜋 3 + 𝑖 2 𝜋 3
  • E1

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