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

Q2:

Write 𝜔 1 1 in its simplest form.

  • A 𝜔
  • B1
  • C 𝜔 2

Q3:

Write 𝜔 1 4 9 in its simplest form.

  • A1
  • B 𝜔 2
  • C 𝜔

Q4:

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

  • A 𝜔
  • B 𝜔
  • C 1 9 , 6 8 3
  • D 1 1 9 , 6 8 3
  • E 19,683

Q5:

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

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

Q6:

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

Q7:

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

  • A 𝜔
  • B1
  • C 𝜔
  • D0

Q8:

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

  • A 𝜔
  • B 1
  • C 𝜔
  • D0

Q9:

What is 6 𝑟 = 0 𝑟 ( 8 + 𝜔 ) ?

  • A0
  • B48
  • C49
  • D57

Q10:

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

Q11:

If 1 + 𝜔 = 𝑎 + 𝑏 𝜔 , 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 2 1 3 3 .

  • A 𝜔
  • B 1
  • C1
  • D 𝜔 2
  • 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 2 2 8 .

  • A1
  • B 𝜔 2
  • C 2 3 2 𝑖
  • D 𝜔
  • E 3 2 3 𝑖

Q16:

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

  • A 1 𝜔
  • B 𝜔 2
  • C 1 𝜔 2
  • D 1 + 𝜔 2

Q17:

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

  • A 𝜔
  • B 𝜔
  • C1
  • D 𝜔 2

Q18:

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

  • A 𝜔
  • B 𝜔
  • C 𝜔
  • D 2 7
  • E1

Q19:

Evaluate 2 + 2 𝜔 + 2 𝜔 where 𝜔 is a complex cube root of unity.

Q20:

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

  • A1
  • B 𝜔
  • C 3 + 1 2 𝑖
  • D 𝜔
  • E 1 2 + 3 2 𝑖

Q21:

Write 𝜔 4 in its simplest form.

  • A 𝜔
  • B1
  • C 𝜔 2

Q22:

Write 𝜔 2 1 in its simplest form.

  • A 𝜔 2
  • B 𝜔
  • C1

Q23:

Write 𝜔 8 5 in its simplest form.

  • A 𝜔
  • B1
  • C 𝜔 2

Q24:

Write 𝜔 9 in its simplest form.

  • A 𝜔 2
  • B 𝜔
  • C1

Q25:

Write 𝜔 1 3 0 in its simplest form.

  • A1
  • B 𝜔 2
  • C 𝜔

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