Lesson Worksheet: Arbitrary Roots of Complex Numbers Mathematics

In this worksheet, we will practice using de Moivre’s theorem to find the 𝑛th roots of a complex number and exploring their properties.

Q1:

Determine the square roots of ๐‘ง, given that ๐‘ง=โˆ’8๐‘–.

  • A{1,โˆ’1}
  • B{๐‘–,โˆ’๐‘–}
  • C๏ฏโˆš32+12๐‘–,โˆ’โˆš32โˆ’12๐‘–๏ป
  • D{2โˆ’2๐‘–,โˆ’2+2๐‘–}
  • E๏ฏ1โˆš2โˆ’1โˆš2๐‘–,โˆ’1โˆš2+1โˆš2๐‘–๏ป

Q2:

Given that ๐‘ง=โˆ’28+96๐‘–, determine the square roots of ๐‘ง without first converting it to trigonometric form.

  • A๏€ผโˆ’725+2425๐‘–๏ˆ,โˆ’๏€ผโˆ’725+2425๐‘–๏ˆ
  • B(6+8๐‘–),โˆ’(6+8๐‘–)
  • C(8+6๐‘–),โˆ’(8+6๐‘–)
  • D๏€ผ2425โˆ’725๐‘–๏ˆ,โˆ’๏€ผ2425โˆ’725๐‘–๏ˆ
  • E๏€ผโˆš2โˆ’12๐‘–๏ˆ,โˆ’๏€ผโˆš2โˆ’12๐‘–๏ˆ

Q3:

Use De Moivreโ€™s theorem to find the two square roots of 16๏€ผ5๐œ‹3โˆ’๐‘–5๐œ‹3๏ˆcossin.

  • Aยฑ4๐‘–
  • Bยฑ๏€ฟ12โˆ’1โˆš2๐‘–๏‹
  • Cยฑ๏€ป2โˆš3+2๐‘–๏‡
  • Dยฑ๏€ฟโˆš32+12๐‘–๏‹

Q4:

Given that ๐‘ง=๏€ป๏€ป๐œ‹6๏‡+๐‘–๏€ป๐œ‹6๏‡๏‡sincos, determine the cubic roots of ๏€น๐‘ง๏…๏Šซ.

  • A๏€ผ๏€ผ5๐œ‹9๏ˆ+๐‘–๏€ผ5๐œ‹9๏ˆ๏ˆcossin, ๏€ผ๏€ผโˆ’7๐œ‹9๏ˆ+๐‘–๏€ผโˆ’7๐œ‹9๏ˆ๏ˆcossin, ๏€ป๏€ปโˆ’๐œ‹9๏‡+๐‘–๏€ปโˆ’๐œ‹9๏‡๏‡cossin
  • B๏€ผ๏€ผ7๐œ‹9๏ˆ+๐‘–๏€ผ7๐œ‹9๏ˆ๏ˆcossin, ๏€ผ๏€ผโˆ’5๐œ‹9๏ˆ+๐‘–๏€ผโˆ’5๐œ‹9๏ˆ๏ˆcossin, ๏€ป๏€ป๐œ‹9๏‡+๐‘–๏€ป๐œ‹9๏‡๏‡cossin
  • C๏€ผ๏€ผ5๐œ‹18๏ˆ+๐‘–๏€ผ5๐œ‹18๏ˆ๏ˆcossin, ๏€ผ๏€ผ17๐œ‹18๏ˆ+๐‘–๏€ผ17๐œ‹18๏ˆ๏ˆcossin, ๏€ผ๏€ผโˆ’7๐œ‹18๏ˆ+๐‘–๏€ผโˆ’7๐œ‹18๏ˆ๏ˆcossin
  • D๏€ผ๏€ผ7๐œ‹9๏ˆ+๐‘–๏€ผ7๐œ‹9๏ˆ๏ˆcossin, ๏€ผ๏€ผโˆ’8๐œ‹9๏ˆ+๐‘–๏€ผโˆ’8๐œ‹9๏ˆ๏ˆcossin, ๏€ผ๏€ผโˆ’5๐œ‹9๏ˆ+๐‘–๏€ผโˆ’5๐œ‹9๏ˆ๏ˆcossin
  • E๏€ผ๏€ผโˆ’7๐œ‹9๏ˆ+๐‘–๏€ผโˆ’7๐œ‹9๏ˆ๏ˆcossin, ๏€ผ๏€ผโˆ’4๐œ‹9๏ˆ+๐‘–๏€ผโˆ’4๐œ‹9๏ˆ๏ˆcossin, ๏€ป๏€ปโˆ’๐œ‹9๏‡+๐‘–๏€ปโˆ’๐œ‹9๏‡๏‡cossin

Q5:

Find the possible real values of 1โˆš3๏€ฝ๐‘–+๐‘–๏‰๏Ž ๏Žข๏Ž ๏Žข๏Šฑ.

  • Aโˆ’2, 0, and 2
  • Bโˆ’1, 0, and 1
  • Cโˆ’2โˆš3, 0, and 2โˆš3
  • Dโˆ’1โˆš3, 0, and 1โˆš3
  • Eโˆ’13, 0, and 13

Q6:

Solve ๐‘ง=16โˆš2+16๐‘–โˆš2๏Šซ.

  • A๐‘ง=2๐‘’๏‘ฝ๏Žก๏ŽŸ๏ƒ, 2๐‘’๏Žจ๏‘ฝ๏Žก๏ŽŸ๏ƒ, 2๐‘’๏Ž ๏Žฆ๏‘ฝ๏Žก๏ŽŸ๏ƒ, 2๐‘’๏Šฑ๏ƒ๏Žฆ๏‘ฝ๏Žก๏ŽŸ, 2๐‘’๏Šฑ๏ƒ๏Žข๏‘ฝ๏Žฃ
  • B๐‘ง=2๐‘’๏‘ฝ๏Ž ๏Žค๏ƒ, 2๐‘’๏Žฆ๏‘ฝ๏Ž ๏Žค๏ƒ, 2๐‘’๏Ž ๏Žข๏‘ฝ๏Ž ๏Žค๏ƒ, 2๐‘’๏Šฑ๏ƒ๏Ž ๏‘ฝ๏Žข, 2๐‘’๏Šฑ๏ƒ๏Ž ๏Ž ๏‘ฝ๏Ž ๏Žค
  • C๐‘ง=4๐‘’๏‘ฝ๏Žก๏ŽŸ๏ƒ, 4๐‘’๏Žจ๏‘ฝ๏Žก๏ŽŸ๏ƒ, 4๐‘’๏Ž ๏Žฆ๏‘ฝ๏Žก๏ŽŸ๏ƒ, 4๐‘’๏Šฑ๏ƒ๏Žฆ๏‘ฝ๏Žก๏ŽŸ, 4๐‘’๏Šฑ๏ƒ๏Žข๏‘ฝ๏Žฃ
  • D๐‘ง=4๐‘’๏‘ฝ๏Ž ๏Žค๏ƒ, 4๐‘’๏Žฆ๏‘ฝ๏Ž ๏Žค๏ƒ, 4๐‘’๏Ž ๏Žข๏‘ฝ๏Ž ๏Žค๏ƒ, 4๐‘’๏Šฑ๏ƒ๏Ž ๏‘ฝ๏Žข, 4๐‘’๏Šฑ๏ƒ๏Ž ๏Ž ๏‘ฝ๏Ž ๏Žค
  • E๐‘ง=32๐‘’๏‘ฝ๏Žฃ๏ƒ, 32๐‘’๏Žจ๏‘ฝ๏Žฃ๏ƒ, 32๐‘’๏Ž ๏Žฆ๏‘ฝ๏Žฃ๏ƒ, 32๐‘’๏Šฑ๏ƒ๏Žฆ๏‘ฝ๏Žฃ, 32๐‘’๏Šฑ๏Šฉ๏Ž„๏ƒ

By plotting these solutions on an Argand diagram, or otherwise, describe the geometric properties of the solutions.

  • AThe roots lie at the vertices of a regular pentagon inscribed in a circle of radius 32 at the origin.
  • BThe roots lie at the vertices of a regular pentagon inscribed in a circle of radius 1 at the origin.
  • CThe roots lie at the vertices of a regular pentagon inscribed in a circle of radius 4 at the origin.
  • DThe roots lie in a straight line.
  • EThe roots lie at the vertices of a regular pentagon inscribed in a circle of radius 2 at the origin.

Q7:

Find the solutions to the equation ๐‘ง=125๐‘’๏Šฌ๏ƒ๏Žก๏‘ฝ๏Žข. What are their geometrical properties?

  • A5๐‘’๏‘ฝ๏Žจ๏ƒ, 5๐‘’๏Žฃ๏‘ฝ๏Žจ๏ƒ, 5๐‘’๏Žฆ๏‘ฝ๏Žจ๏ƒ, 5๐‘’๏Šฑ๏ƒ๏Žก๏‘ฝ๏Žจ, 5๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žจ, 5๐‘’๏Šฑ๏ƒ๏Žง๏‘ฝ๏Žจ; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius 5.
  • Bโˆš5๐‘’๏‘ฝ๏Žจ๏ƒ, โˆš5๐‘’๏Žฃ๏‘ฝ๏Žจ๏ƒ, โˆš5๐‘’๏Žฆ๏‘ฝ๏Žจ๏ƒ, โˆš5๐‘’๏Šฑ๏ƒ๏Žก๏‘ฝ๏Žจ, โˆš5๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žจ, โˆš5๐‘’๏Šฑ๏ƒ๏Žง๏‘ฝ๏Žจ; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius โˆš5.
  • C5๐‘’๏‘ฝ๏Žจ๏ƒ, 5๐‘’๏Žฃ๏‘ฝ๏Žจ๏ƒ, 5๐‘’๏Žฆ๏‘ฝ๏Žจ๏ƒ, 5๐‘’๏Šฑ๏ƒ๏Žก๏‘ฝ๏Žจ, 5๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žจ, 5๐‘’๏Šฑ๏ƒ๏Žง๏‘ฝ๏Žจ; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius โˆš5.
  • Dโˆš5๐‘’๏‘ฝ๏Žจ๏ƒ, โˆš5๐‘’๏Žค๏‘ฝ๏Ž ๏Žง๏ƒ, โˆš5๐‘’๏Žฃ๏‘ฝ๏Žจ๏ƒ, โˆš5๐‘’๏Ž ๏Ž ๏‘ฝ๏Ž ๏Žง๏ƒ, โˆš5๐‘’๏Žฆ๏‘ฝ๏Žจ๏ƒ, โˆš5๐‘’๏Ž ๏Žฆ๏‘ฝ๏Ž ๏Žง๏ƒ; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius โˆš5.
  • E125๐‘’๏Žก๏‘ฝ๏Žข๏ƒ, 125๐‘’๏Žง๏‘ฝ๏Žข๏ƒ, 125๐‘’๏Žฆ๏‘ฝ๏Žจ๏ƒ, 125๐‘’๏Šฑ๏ƒ๏Ž ๏Žฃ๏‘ฝ๏Žข, 125๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žข, 125๐‘’๏Šฑ๏ƒ๏Žง๏‘ฝ๏Žจ; the roots lie at a straight line that passes through the origin.

State the 6th roots of unity.

  • A๐‘’๏‘ฝ๏Žฅ๏ƒ, ๐‘–, ๐‘’๏Žค๏‘ฝ๏Žฅ๏ƒ, ๐‘’๏Šฑ๏ƒ๏‘ฝ๏Žฅ, โˆ’๐‘–, ๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žฅ
  • B1, ๐‘’๏‘ฝ๏Žข๏ƒ, ๐‘–, โˆ’1, ๐‘’๏Šฑ๏ƒ๏‘ฝ๏Žข, โˆ’๐‘–
  • C1, ๐‘’๏‘ฝ๏Žฅ๏ƒ, ๐‘’๏‘ฝ๏Žข๏ƒ, ๐‘’๏‘ฝ๏Žก๏ƒ, ๐‘’๏Žก๏‘ฝ๏Žข๏ƒ, ๐‘’๏Žค๏‘ฝ๏Žฅ๏ƒ
  • D1, ๐‘’๏‘ฝ๏Žฅ๏ƒ, ๐‘’๏Žค๏‘ฝ๏Žฅ๏ƒ, โˆ’1, ๐‘’๏Šฑ๏ƒ๏‘ฝ๏Žฅ, ๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žฅ
  • E1, ๐‘’๏‘ฝ๏Žข๏ƒ, ๐‘’๏Žก๏‘ฝ๏Žข๏ƒ, โˆ’1, ๐‘’๏Šฑ๏ƒ๏‘ฝ๏Žข, ๐‘’๏Šฑ๏ƒ๏Žก๏‘ฝ๏Žข

What is the relationship between the 6th roots of unity and the solutions to the equation ๐‘ง=125๐‘’๏Šฌ๏ƒ๏Žก๏‘ฝ๏Žข?

  • AThe solutions to the equation are the 6th roots of unity multiplied by 5๐‘’๏‘ฝ๏Žจ๏ƒ.
  • BThe solutions to the equation are the 6th roots of unity multiplied by 125๐‘’๏Žก๏‘ฝ๏Žข๏ƒ.
  • CThe solutions to the equation are the 6th roots of unity multiplied by 5๐‘’๏‘ฝ๏Žข.
  • DThe solutions to the equation are the 6th roots of unity multiplied by โˆš5๐‘’๏‘ฝ๏Žจ.
  • EThe solutions to the equation are the 6th roots of unity multiplied by โˆš5๐‘’๏‘ฝ๏Žจ๏ƒ.

Q8:

Find the roots of ๐‘ง+16=0๏Šฎ.

  • Aโˆš2๐‘’๏‘ฝ๏Žง๏ƒ, โˆš2๐‘’๏Žข๏‘ฝ๏Žง๏ƒ, โˆš2๐‘’๏Žค๏‘ฝ๏Žง๏ƒ, โˆš2๐‘’๏Žฆ๏‘ฝ๏Žง๏ƒ, โˆš2๐‘’๏Šฑ๏ƒ๏Žฆ๏‘ฝ๏Žง, โˆš2๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žง, โˆš2๐‘’๏Šฑ๏ƒ๏Žข๏‘ฝ๏Žง, โˆš2๐‘’๏Šฑ๏ƒ๏‘ฝ๏Žง.
  • Bโˆš2๐‘’๏‘ฝ๏Žฃ๏ƒ, โˆš2๐‘’๏Žข๏‘ฝ๏Žฃ๏ƒ, โˆš2๐‘’๏Žค๏‘ฝ๏Žฃ๏ƒ, โˆš2๐‘’๏Žฆ๏‘ฝ๏Žฃ๏ƒ, โˆš2๐‘’๏Šฑ๏ƒ๏Žฆ๏‘ฝ๏Žฃ, โˆš2๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žฃ, โˆš2๐‘’๏Šฑ๏ƒ๏Žข๏‘ฝ๏Žฃ, โˆš2๐‘’๏Šฑ๏ƒ๏‘ฝ๏Žฃ
  • C2๐‘’๏‘ฝ๏Žง๏ƒ, 2๐‘’๏Žข๏‘ฝ๏Žง๏ƒ, 2๐‘’๏Žค๏‘ฝ๏Žง๏ƒ, 2๐‘’๏Žฆ๏‘ฝ๏Žง๏ƒ, 2๐‘’๏Šฑ๏ƒ๏Žฆ๏‘ฝ๏Žง, 2๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žง, 2๐‘’๏Šฑ๏ƒ๏Žข๏‘ฝ๏Žง, 2๐‘’๏Šฑ๏ƒ๏‘ฝ๏Žง
  • D๐‘’๏‘ฝ๏Ž ๏Žฅ, ๐‘’๏Žข๏‘ฝ๏Ž ๏Žฅ, ๐‘’๏Žค๏‘ฝ๏Ž ๏Žฅ, ๐‘’๏Žฆ๏‘ฝ๏Ž ๏Žฅ, ๐‘’๏Žจ๏‘ฝ๏Ž ๏Žฅ, ๐‘’๏Ž ๏Ž ๏‘ฝ๏Ž ๏Žฅ, ๐‘’๏Ž ๏Žข๏‘ฝ๏Ž ๏Žฅ, ๐‘’๏Ž ๏Žค๏‘ฝ๏Ž ๏Žฅ
  • E2๐‘’๏‘ฝ๏Žฃ๏ƒ, 2๐‘’๏Žข๏‘ฝ๏Žฃ๏ƒ, 2๐‘’๏Žค๏‘ฝ๏Žฃ๏ƒ, 2๐‘’๏Žฆ๏‘ฝ๏Žฃ๏ƒ, 2๐‘’๏Šฑ๏ƒ๏Žฆ๏‘ฝ๏Žฃ, 2๐‘’๏Šฑ๏ƒ๏Žค๏‘ฝ๏Žฃ, 2๐‘’๏Šฑ๏ƒ๏Žข๏‘ฝ๏Žฃ, 2๐‘’๏Šฑ๏ƒ๏‘ฝ๏Žฃ

The complex numbers representing the roots of ๐‘ง+16=0๏Šฎ are each squared to form the vertices of a new shape. What is the area of this shape?

Q9:

Find the coordinates of the vertices of a regular hexagon centered at (โˆ’1,2) with one vertex at the origin.

Give your answer as exact Cartesian coordinates.

  • A(0,0), ๏€ฟ3+2โˆš32,โˆš3โˆ’62๏‹, ๏€ฟ1+2โˆš32,โˆš3โˆ’22๏‹, (2,โˆ’4), ๏€ฟ1โˆ’2โˆš32,โˆ’โˆš3โˆ’22๏‹, ๏€ฟ3โˆ’2โˆš32,โˆ’โˆš3โˆ’62๏‹
  • B(0,0), ๏€ฟโˆ’2โˆ’โˆš32,โˆ’2โˆš3+12๏‹, ๏€ฟโˆ’6โˆ’โˆš32,โˆ’2โˆš3+12๏‹, (โˆ’4,2), ๏€ฟโˆ’6+โˆš32,2โˆš3+32๏‹, ๏€ฟโˆ’2+โˆš32,2โˆš3+12๏‹
  • C(0,0), ๏€ฟโˆ’1+2โˆš32,โˆš3+22๏‹, ๏€ฟโˆ’3+2โˆš32,โˆš3+62๏‹, (โˆ’2,4), ๏€ฟโˆ’3โˆ’2โˆš32,โˆ’โˆš3+62๏‹, ๏€ฟโˆ’1โˆ’2โˆš32,โˆ’โˆš3+22๏‹
  • D(0,0), (โˆ’2,0), (โˆ’3,2), (โˆ’2,4), (0,4), (1,2)
  • E(0,0), ๏€ฟ2+โˆš32,2โˆš3โˆ’12๏‹, ๏€ฟ6+โˆš32,2โˆš3โˆ’32๏‹, (4,โˆ’2), ๏€ฟ6โˆ’โˆš32,โˆ’2โˆš3โˆ’32๏‹, ๏€ฟ2โˆ’โˆš32,โˆ’2โˆš3โˆ’12๏‹

Q10:

Find the coordinates of the vertices of a regular pentagon centered at the origin with one vertex at (3,3).

Give your answer as exact Cartesian coordinates.

  • A(3,3), ๏€ผ3โˆš2๏€ผ13๐œ‹20๏ˆ,3โˆš2๏€ผ13๐œ‹20๏ˆ๏ˆsincos, ๏€ผ3โˆš2๏€ผโˆ’19๐œ‹20๏ˆ,3โˆš2๏€ผโˆ’19๐œ‹20๏ˆ๏ˆsincos, ๏€ผ3โˆš2๏€ผโˆ’11๐œ‹20๏ˆ,3โˆš2๏€ผโˆ’11๐œ‹20๏ˆ๏ˆsincos, ๏€ผ3โˆš2๏€ผโˆ’3๐œ‹20๏ˆ,3โˆš2๏€ผโˆ’3๐œ‹20๏ˆ๏ˆsincos
  • B(3,3), ๏€ผ3โˆš2๏€ผ13๐œ‹20๏ˆ,3โˆš2๏€ผ13๐œ‹20๏ˆ๏ˆcossin, ๏€ผ3โˆš2๏€ผโˆ’19๐œ‹20๏ˆ,3โˆš2๏€ผโˆ’19๐œ‹20๏ˆ๏ˆcossin, ๏€ผ3โˆš2๏€ผโˆ’11๐œ‹20๏ˆ,3โˆš2๏€ผโˆ’11๐œ‹20๏ˆ๏ˆcossin, ๏€ผ3โˆš2๏€ผโˆ’3๐œ‹20๏ˆ,3โˆš2๏€ผโˆ’3๐œ‹20๏ˆ๏ˆcossin
  • C(3,3), ๏€ผ3โˆš2๏€ผ11๐œ‹15๏ˆ,3โˆš2๏€ผ11๐œ‹15๏ˆ๏ˆcossin, ๏€ผ3โˆš2๏€ผโˆ’13๐œ‹15๏ˆ,3โˆš2๏€ผโˆ’13๐œ‹15๏ˆ๏ˆcossin, ๏€ผ3โˆš2๏€ผโˆ’7๐œ‹15๏ˆ,3โˆš2๏€ผโˆ’7๐œ‹15๏ˆ๏ˆcossin, ๏€ป3โˆš2๏€ปโˆ’๐œ‹15๏‡,3โˆš2๏€ปโˆ’๐œ‹15๏‡๏‡cossin
  • D(3,3), ๏€ผ3๏€ผ13๐œ‹20๏ˆ,3๏€ผ13๐œ‹20๏ˆ๏ˆsincos, ๏€ผ3๏€ผโˆ’19๐œ‹20๏ˆ,3๏€ผโˆ’19๐œ‹20๏ˆ๏ˆsincos, ๏€ผ3๏€ผโˆ’11๐œ‹20๏ˆ,3๏€ผโˆ’11๐œ‹20๏ˆ๏ˆsincos, ๏€ผ3๏€ผโˆ’3๐œ‹20๏ˆ,3๏€ผโˆ’3๐œ‹20๏ˆ๏ˆsincos
  • E(3,3), ๏€ผ3๏€ผ13๐œ‹20๏ˆ,3๏€ผ13๐œ‹20๏ˆ๏ˆcossin, ๏€ผ3๏€ผโˆ’19๐œ‹20๏ˆ,3๏€ผโˆ’19๐œ‹20๏ˆ๏ˆcossin, ๏€ผ3๏€ผโˆ’11๐œ‹20๏ˆ,3๏€ผโˆ’11๐œ‹20๏ˆ๏ˆcossin, ๏€ผ3๏€ผโˆ’3๐œ‹20๏ˆ,3๏€ผโˆ’3๐œ‹20๏ˆ๏ˆcossin

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