Worksheet: Cube Roots of Unity

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

Q1:

If 𝑧=𝜔, where 𝑥 is a positive integer and 𝜔is one of the complex cubic roots of unity, then what is |𝑧|?

Q2:

Write 𝜔 in its simplest form.

  • A 𝜔
  • B 𝜔
  • C1

Q3:

Write 𝜔 in its simplest form.

  • A 𝜔
  • B 𝜔
  • C1

Q4:

Evaluate (1𝜔), where 𝜔 is a primitive cubic root of unity.

  • A19,683
  • B 𝜔
  • C 1 1 9 , 6 8 3
  • D 𝜔
  • E 1 9 , 6 8 3

Q5:

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

  • A 𝜔
  • B 1 2 2 𝑖
  • C 3 3 2 𝑖
  • D1
  • E 𝜔

Q6:

Evaluate 66𝜔6𝜔 where 𝜔 is a complex cube root of unity.

Q7:

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

  • A 𝜔
  • B 𝜔
  • C1
  • D0

Q8:

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

  • A 𝜔
  • B 𝜔
  • C0
  • D 1

Q9:

What is (8+𝜔)?

  • A49
  • B48
  • C0
  • D57

Q10:

Evaluate 7+𝜔+𝜔 where 𝜔 is a complex cube root of unity.

Q11:

If 1+𝜔=𝑎+𝑏𝜔, where 𝑎 and 𝑏 are real numbers and 𝜔 is a complex cube root of unity, then what is (𝑎,𝑏)?

  • A ( 0 , 0 )
  • B ( 1 , 0 )
  • C ( 1 , 1 )
  • D ( 1 , 1 )

Q12:

Evaluate 1+𝜔.

  • A1
  • B 1
  • C 𝜔
  • D 𝜔
  • E 𝜔

Q13:

What is the smallest positive integer value of 𝑛 for which 1+𝜔=(1+𝜔) where 𝜔 is a complex cube root of unity?

Q14:

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

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

Q15:

Evaluate 1+𝜔.

  • A 𝜔
  • B 3 2 3 𝑖
  • C1
  • D 𝜔
  • E 2 3 2 𝑖

Q16:

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

  • A 1 𝜔
  • B 1 + 𝜔
  • C 𝜔
  • D 1 𝜔

Q17:

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

  • A 𝜔
  • B1
  • C 𝜔
  • D 𝜔

Q18:

Evaluate (1𝜔), where 𝜔 is a primitive cubic root of unity.

  • A 𝜔
  • B1
  • C 𝜔
  • D 2 7
  • E 𝜔

Q19:

Evaluate 6+6𝜔+6𝜔 where 𝜔 is a complex cube root of unity.

Q20:

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

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

Q21:

Write 𝜔 in its simplest form.

  • A 𝜔
  • B1
  • C 𝜔

Q22:

Write 𝜔 in its simplest form.

  • A 𝜔
  • B1
  • C 𝜔

Q23:

Write 𝜔 in its simplest form.

  • A 𝜔
  • B1
  • C 𝜔

Q24:

Write 𝜔 in its simplest form.

  • A1
  • B 𝜔
  • C 𝜔

Q25:

Write 𝜔 in its simplest form.

  • A 𝜔
  • B1
  • C 𝜔

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