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 ๐‘ง=1๏Š in exponential form?

  • A๐‘’๏Žก๏‘ฝ๏‘ƒ๏‘€๏ƒ
  • B๐‘’๏Ž„๏‡๏Š๏ƒ
  • C๐‘›๐‘’๏Šจ๏Ž„๏‡๏ƒ
  • D๐‘’๏Žก๏‘ฝ๏‘€๏‘ƒ๏ƒ
  • E๐‘’๏Šจ๏Ž„๏‡๏Š๏ƒ

Q2:

Let ๐œ” be an ๐‘›th root of unity. When can we define ๐œ” as a primitive ๐‘›th root of unity?

  • AWhen it is an ๐‘šth root of unity for some ๐‘š<๐‘›
  • BWhen it is an ๐‘šth root of unity, where ๐‘š๐‘› is a prime number
  • COnly when ๐‘› is a prime number
  • DOnly when ๐‘› is an even number
  • EWhen it is not an ๐‘šth root of unity for some ๐‘š<๐‘›

Q3:

If ๐œ” is a primitive 6th root of unity, which of the following expressions is equivalent to ๐œ”+๐œ”+๐œ”๏Šจ๏Šฉ?

  • A1โˆ’๐œ”โˆ’๐œ”๏Šช๏Šซ
  • B1
  • C๐œ”+๐œ”+๐œ”๏Šช๏Šซ๏Šฌ
  • D12๏€น๐œ”+๐œ”+๐œ”๏…๏Šจ๏Šช๏Šฌ
  • Eโˆ’๏€น1+๐œ”+๐œ”๏…๏Šช๏Šซ

Q4:

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

  • Aโˆš2โˆ’โˆš2๐‘–
  • Bโˆ’12โˆ’โˆš32๐‘–
  • Cโˆš32โˆ’12๐‘–
  • D2โˆš2+2โˆš2๐‘–
  • Eโˆš22โˆ’โˆš22๐‘–

Q5:

Let ๐œ” be one of the quintic roots of unity. Which of the following is an equivalent expression to ๐œ”๏Šฑ๏Šฉ?

  • A๐œ”๏Šจ
  • B1๐œ”๏Šจ
  • Cโˆ’๐œ”๏Šฉ
  • D๐œ”๏Šฎ
  • E๐œ”๏Šฑ๏Šง๏Šซ

Q6:

Let ๐œ” be an ๐‘›th root of unity and ๐‘˜ be a positive integer. Which of the following is not an equivalent expression for ๐œ”๏Šฑ๏‡?

  • A๏€น๐œ”๏…๏‡โˆ—
  • B๐œ”๏Š๏Šฑ๏‡
  • C๐œ”๏Š๏Šฐ๏‡
  • D1๐œ”๏‡
  • E๏€น๐œ”๏…๏‡๏Šฑ๏Šง

Q7:

Let ๐œ” be an ๐‘›th root of unity, where ๐‘› is even. Which of the following expressions is equivalent to โˆ’๐œ”๏‡?

  • A๐œ”๏‡๏Šฐ๏‘ƒ๏Žฃ
  • B๐œ”๏‡๏Šฐ๏Š
  • C๐œ”๏‡๏Šฐ๏‘ƒ๏Žก
  • D(๐œ”)๏Šฑ๏‡โˆ—
  • E๐œ”๏Šฑ๏‡

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((2๐œ‹๐‘˜)+๐‘–(2๐œ‹๐‘˜))cossin๏Šง๏Šฆ
  • Bcossin๏€ฝ10๐œ‹๐‘˜2๏‰+๐‘–๏€ฝ10๐œ‹๐‘˜2๏‰
  • Ccossin(10๐œ‹๐‘˜)+๐‘–(10๐œ‹๐‘˜)
  • Dsincos๏€ฝ2๐œ‹๐‘˜10๏‰+๐‘–๏€ฝ2๐œ‹๐‘˜10๏‰
  • Ecossin๏€ฝ2๐œ‹๐‘˜10๏‰+๐‘–๏€ฝ2๐œ‹๐‘˜10๏‰

Using the general form for 10th roots of unity, identify the 10th root of unity for the case where ๐‘˜=3.

  • Acossin(30๐œ‹)+๐‘–(30๐œ‹)
  • Bcossin๏€ผโˆ’3๐œ‹5๏ˆ+๐‘–๏€ผโˆ’3๐œ‹5๏ˆ
  • Ccossin๏€ผโˆ’3๐œ‹10๏ˆ+๐‘–๏€ผโˆ’3๐œ‹10๏ˆ
  • Dcossin๏€ผ3๐œ‹5๏ˆ+๐‘–๏€ผ3๐œ‹5๏ˆ
  • Ecossin๏€ผ3๐œ‹5๏ˆโˆ’๐‘–๏€ผ3๐œ‹5๏ˆ

Q10:

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

  • A1
  • Bcossin๏€ผ2๐œ‹3๏ˆ+๐‘–๏€ผ2๐œ‹3๏ˆ
  • Ccossin๏€ผ2๐œ‹3๏ˆโˆ’๐‘–๏€ผ2๐œ‹3๏ˆ
  • Dcossin๏€ผโˆ’2๐œ‹3๏ˆ+๐‘–๏€ผโˆ’2๐œ‹3๏ˆ
  • Ecossin๏€ป๐œ‹3๏‡+๐‘–๏€ป๐œ‹3๏‡

Q11:

Write a general form for the roots ๐‘ง=1๏Š, giving your answer in polar form.

  • Acossin๏€ฝ2๐œ‹๐‘˜๐‘›๏‰+๐‘–๏€ฝ2๐œ‹๐‘˜๐‘›๏‰
  • Bsincos๏€ฝ2๐œ‹๐‘˜๐‘›๏‰+๐‘–๏€ฝ2๐œ‹๐‘˜๐‘›๏‰
  • Ccossin(2๐œ‹๐‘˜๐‘›)+๐‘–(2๐œ‹๐‘˜๐‘›)
  • Dcossin๏€ผ2๐œ‹๐‘›๐‘˜๏ˆ+๐‘–๏€ผ2๐œ‹๐‘›๐‘˜๏ˆ
  • E((2๐œ‹๐‘˜)+๐‘–(2๐œ‹๐‘˜))cossin๏Š

Q12:

Find the quintic roots of unity.

  • A1, ๐‘’๏Žก๏‘ฝ๏Žค๏ƒ, ๐‘’๏Žฃ๏‘ฝ๏Žค๏ƒ, ๐‘’๏Šฑ๏ƒ๏Žฃ๏‘ฝ๏Žค, ๐‘’๏Šฑ๏ƒ๏Žก๏‘ฝ๏Žค
  • Bโˆ’1, ๐‘’๏Šซ๏Ž„๏ƒ, ๐‘’๏Žค๏‘ฝ๏Žก๏ƒ, ๐‘’๏Žค๏‘ฝ๏Žข๏ƒ, ๐‘’๏Žค๏‘ฝ๏Žฃ๏ƒ
  • C1, ๐‘’๏‘ฝ๏Žค๏ƒ, ๐‘’๏Žก๏‘ฝ๏Žค๏ƒ, ๐‘’๏Žข๏‘ฝ๏Žค๏ƒ, ๐‘’๏Žฃ๏‘ฝ๏Žค๏ƒ
  • D1, ๐‘’๏Šซ๏Ž„๏ƒ, ๐‘’๏Šง๏Šฆ๏Ž„๏ƒ, ๐‘’๏Šง๏Šซ๏Ž„๏ƒ, ๐‘’๏Šจ๏Šฆ๏Ž„๏ƒ
  • Eโˆ’1, ๐‘’๏Žก๏‘ฝ๏Žค๏ƒ, ๐‘’๏Žฃ๏‘ฝ๏Žค๏ƒ, ๐‘’๏Žฅ๏‘ฝ๏Žค๏ƒ, ๐‘’๏Žง๏‘ฝ๏Žค๏ƒ

What is the value of their sum?

Q13:

Let ๐œ” be an ๐‘›th root of unity.

Which of the following is the correct relationship between ๐œ”๏Šฑ๏Šง and ๐œ”?

  • A๐œ”=โˆ’๐œ”๏Šฑ๏Šง
  • B๐œ”=(๐œ”)๏Šฑ๏Šงโˆ—
  • C๐œ”=๐œ”๏Šฑ๏Šง
  • D๐œ”=โˆ’(๐œ”)๏Šฑ๏Šงโˆ—

Express ๐œ”๏Šฑ๏Šง in terms of positive powers of ๐œ”.

  • A๐œ”
  • B๐œ”๏Š๏Šฑ๏Šง
  • Cโˆ’๐œ”
  • D๐œ”๏Š๏Šฐ๏Šง
  • E๐œ”๏‘ƒ๏Žก

Q14:

For how many pairs of real numbers (๐‘Ž,๐‘) does the relation (๐‘Ž+๐‘๐‘–)=๐‘Žโˆ’๐‘๐‘–๏Šจ๏Ž•๏Šฆ๏Šจ๏Šฆ hold?

Q15:

Find the cube roots of unity.

  • A1, cossin(๐œ‹)+๐‘–(๐œ‹), cossin(2๐œ‹)+๐‘–(2๐œ‹)
  • B1, cossin๏€ป๐œ‹3๏‡+๐‘–๏€ป๐œ‹3๏‡, cossin๏€ปโˆ’๐œ‹3๏‡+๐‘–๏€ปโˆ’๐œ‹3๏‡
  • C1, sincos๏€ผ2๐œ‹3๏ˆ+๐‘–@๏€ผ2๐œ‹3๏ˆ, sincos๏€ผโˆ’2๐œ‹3๏ˆ+๐‘–๏€ผโˆ’2๐œ‹3๏ˆ
  • D1, cossin๏€ผ2๐œ‹3๏ˆโˆ’๐‘–๏€ผ2๐œ‹3๏ˆ, cossin๏€ผโˆ’2๐œ‹3๏ˆโˆ’๐‘–๏€ผโˆ’2๐œ‹3๏ˆ
  • E1, cossin๏€ผ2๐œ‹3๏ˆ+๐‘–๏€ผ2๐œ‹3๏ˆ, cossin๏€ผโˆ’2๐œ‹3๏ˆ+๐‘–๏€ผโˆ’2๐œ‹3๏ˆ

Find the solutions to ๐‘ง=1๏Šฌ.

  • A1, cossin๏€ผ2๐œ‹6๏ˆ+๐‘–๏€ผ2๐œ‹6๏ˆ, cossin๏€ผ2๐œ‹3๏ˆ+๐‘–๏€ผ2๐œ‹3๏ˆ, โˆ’1, cossin๏€ผโˆ’2๐œ‹3๏ˆ+๐‘–๏€ผโˆ’2๐œ‹3๏ˆ, cossin๏€ผโˆ’2๐œ‹6๏ˆ+๐‘–๏€ผโˆ’2๐œ‹6๏ˆ
  • B1, cossin(๐œ‹)โˆ’๐‘–(๐œ‹), cossin(2๐œ‹)โˆ’๐‘–(2๐œ‹), โˆ’1, cossin(4๐œ‹)+๐‘–(4๐œ‹), cossin(5๐œ‹)+๐‘–(5๐œ‹)
  • C1, cossin๏€ป๐œ‹6๏‡โˆ’๐‘–๏€ป๐œ‹6๏‡, cossin๏€ป๐œ‹3๏‡โˆ’๐‘–๏€ป๐œ‹3๏‡, โˆ’1, cossin๏€ปโˆ’๐œ‹3๏‡+๐‘–๏€ปโˆ’๐œ‹3๏‡, cossin๏€ปโˆ’๐œ‹6๏‡+๐‘–๏€ปโˆ’๐œ‹6๏‡
  • D1, cossin๏€ป๐œ‹6๏‡+๐‘–๏€ป๐œ‹6๏‡, cossin๏€ป๐œ‹3๏‡+๐‘–๏€ป๐œ‹3๏‡, โˆ’1, cossin๏€ปโˆ’๐œ‹3๏‡+๐‘–๏€ปโˆ’๐œ‹3๏‡, cossin๏€ปโˆ’๐œ‹6๏‡+๐‘–๏€ปโˆ’๐œ‹6๏‡
  • E1, cossin๏€ผ2๐œ‹6๏ˆโˆ’๐‘–๏€ผ2๐œ‹6๏ˆ, cossin๏€ผ2๐œ‹3๏ˆโˆ’๐‘–๏€ผ2๐œ‹3๏ˆ, โˆ’1, cossin๏€ผโˆ’2๐œ‹3๏ˆ+๐‘–๏€ผโˆ’2๐œ‹3๏ˆ, cossin๏€ผโˆ’2๐œ‹6๏ˆ+๐‘–๏€ผโˆ’2๐œ‹6๏ˆ

What is the relationship between the cubic roots of unity and the 6th roots of unity?

  • A1 is the only common root between the cubic roots of unity and the 6th roots of unity.
  • BAll of the cubic roots of unity are also 6th roots of unity.
  • CAll of the cubic roots of unity and their complex conjugates are 6th roots of unity.
  • DThe cubic roots of unity divided by 2 are 6th roots of unity.
  • EThere are no common roots between the cubic roots of unity and the 6th roots of unity.

Q16:

Two regular polygons are inscribed in the same circle: the first has 1,731 sides and the second has 4,039. If the two polygons have at least one vertex in common, how many vertices in total will coincide?

Q17:

Two regular polygons are inscribed in the same circle where the first has 1,731 sides and the second has 4,039. If the two polygons have at least one vertex in common, how many vertices in total will coincide?

Q18:

Find the cube roots of unity in algebraic form.

  • Aโˆ’1, 12+๐‘–โˆš32, 12โˆ’๐‘–โˆš32
  • Bโˆ’๐‘–, 12+๐‘–โˆš32, โˆ’12+๐‘–โˆš32
  • C1, 12+๐‘–โˆš32, 12โˆ’๐‘–โˆš32
  • D1, โˆ’12+๐‘–โˆš32, โˆ’12โˆ’๐‘–โˆš32
  • E๐‘–, 12โˆ’๐‘–โˆš32, โˆ’12โˆ’๐‘–โˆš32

Plot the roots on an Argand diagram.

  • A
  • B
  • C
  • D
  • E

Q19:

What is the relationship between the cube roots of unity and the sixth roots of unity?

  • ASixth roots of unity are double cube roots of unity.
  • BThere is no relationship between sixth roots of unity and cube roots of unity.
  • CAll sixth roots of unity are also cube roots of unity.
  • DCube roots of unity are half sixth roots of unity.
  • EAll cube roots of unity are also sixth roots of unity.

Q20:

Find the sum of the sixth roots of unity.

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