Worksheet: Cube Roots of Unity

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

Q1:

If ๐‘ง=๐œ”๏—, where ๐‘ฅ is a positive integer and ๐œ”is one of the complex cubic roots of unity, then what is |๐‘ง|?

Q2:

Write ๐œ”๏Šง๏Šง in its simplest form.

  • A๐œ”๏Šจ
  • B๐œ”
  • C1

Q3:

Write ๐œ”๏Šฑ๏Šง๏Šช๏Šฏ in its simplest form.

  • A๐œ”
  • B๐œ”๏Šจ
  • C1

Q4:

Evaluate (1โˆ’๐œ”)๏Šฑ๏Šง๏Šฎ, where ๐œ” is a primitive cubic root of unity.

  • A19,683
  • Bโˆ’๐œ”
  • Cโˆ’119,683
  • Dโˆ’๐œ”๏Šจ
  • Eโˆ’19,683

Q5:

Evaluate ๏€น1+๐œ”๏…๏Šจ๏Šฑ๏Šฌ๏Šฆ, where ๐œ” is a primitive cubic root of unity.

  • A๐œ”
  • B12โˆ’โˆš2๐‘–
  • Cโˆ’โˆš3โˆ’โˆš32๐‘–
  • D1
  • E๐œ”๏Šจ

Q6:

Evaluate โˆ’6โˆ’6๐œ”โˆ’6๐œ”๏Šจ where ๐œ” is a complex cube root of unity.

Q7:

Evaluate ๏Šฎ๏Ž๏Šฒ๏Šฆ๏Ž๏„š๐œ” where ๐œ” is a complex cube root of unity.

  • Aโˆ’๐œ”๏Šจ
  • B๐œ”
  • C1
  • D0

Q8:

Evaluate ๐œ”+๐œ”+๐œ”+๐œ”+โ‹ฏ+๐œ”๏Šจ๏Šฉ๏Šช๏Šง๏Šฎ where ๐œ” is a complex cube root of unity.

  • A๐œ”
  • Bโˆ’๐œ”๏Šจ
  • C0
  • Dโˆ’1

Q9:

What is ๏Šฌ๏Ž๏Šฒ๏Šฆ๏Ž๏„š(8+๐œ”)?

  • A49
  • B48
  • C0
  • D57

Q10:

Evaluate ๏Šซ๏Ž๏Šฒ๏Šฆ๏Ž๏Šจ๏Ž๏„š๏€น7+๐œ”+๐œ”๏… where ๐œ” is a complex cube root of unity.

Q11:

If ๏€น1+๐œ”๏…=๐‘Ž+๐‘๐œ”๏Šจ๏Šง๏Šง, where ๐‘Ž and ๐‘ are real numbers and ๐œ” is a complex cube root of unity, then what is (๐‘Ž,๐‘)?

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

Q12:

Evaluate ๏€น1+๐œ”๏…๏Šจ๏Šฑ๏Šง๏Šฉ๏Šฉ.

  • A1
  • Bโˆ’1
  • Cโˆ’๐œ”๏Šจ
  • D๐œ”
  • Eโˆ’๐œ”

Q13:

What is the smallest positive integer value of ๐‘› for which ๏€น1+๐œ”๏…=(1+๐œ”)๏Šจ๏‘ƒ๏Žฃ๏‘ƒ๏Žฃ where ๐œ” is a complex cube root of unity?

Q14:

Evaluate ๏€น1+๐œ”๏…๏Šจ๏Šฉ๏Šซ, where ๐œ” is a complex cube root of unity.

  • Aโˆ’1
  • Bโˆ’๐œ”
  • Cโˆ’๐œ”๏Šจ
  • D๐œ”๏Šจ
  • E๐œ”

Q15:

Evaluate ๏€น1+๐œ”๏…๏Šจ๏Šจ๏Šฎ.

  • A๐œ”๏Šจ
  • Bโˆ’โˆš32โˆ’โˆš3๐‘–
  • C1
  • D๐œ”
  • Eโˆ’โˆš2โˆ’โˆš32๐‘–

Q16:

Given that ๐œ”is one of the complex cubic roots of unity, find the complex conjugate of 1+๐œ”.

  • A1โˆ’๐œ”๏Šจ
  • B1+๐œ”๏Šจ
  • Cโˆ’๐œ”๏Šจ
  • D1โˆ’๐œ”

Q17:

Given that ๐œ”is one of the complex cubic roots of unity, find the complex conjugate of ๐œ”.

  • A๐œ”๏Šจ
  • B1
  • Cโˆ’๐œ”
  • D๐œ”

Q18:

Evaluate (1โˆ’๐œ”)๏Šฌ, where ๐œ” is a primitive cubic root of unity.

  • A๐œ”๏Šจ
  • B1
  • Cโˆ’๐œ”๏Šจ
  • Dโˆ’27
  • Eโˆ’๐œ”

Q19:

Evaluate 6+6๐œ”+6๐œ”๏Šจ where ๐œ” is a complex cube root of unity.

Q20:

Evaluate ๏€น1+๐œ”๏…๏Šจ๏Šฑ๏Šฏ๏Šฆ, where ๐œ” is a primitive cubic root of unity.

  • A๐œ”
  • B1โˆš2โˆ’12๐‘–
  • Cโˆ’โˆš2+โˆš2๐‘–
  • D1
  • E๐œ”๏Šจ

Q21:

Write ๐œ”๏Šฑ๏Šง๏Šจ๏Šง in its simplest form.

  • A๐œ”๏Šจ
  • B1
  • C๐œ”

Q22:

Write ๐œ”๏Šฑ๏Šง๏Šฉ๏Šฆ in its simplest form.

  • A๐œ”๏Šจ
  • B1
  • C๐œ”

Q23:

Write ๐œ”๏Šฑ๏Šช in its simplest form.

  • A๐œ”๏Šจ
  • B1
  • C๐œ”

Q24:

Write ๐œ”๏Šฏ in its simplest form.

  • A1
  • B๐œ”๏Šจ
  • C๐œ”

Q25:

Write ๐œ”๏Šง๏Šฉ๏Šฆ in its simplest form.

  • A๐œ”
  • B1
  • C๐œ”๏Šจ

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