Lesson Worksheet: Arbitrary Roots of Complex Numbers Mathematics

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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 𝑧, given that 𝑧=−8𝑖.

  • A{1,−1}
  • B{𝑖,−𝑖}
  • C√32+12𝑖,−√32−12𝑖
  • D{2−2𝑖,−2+2𝑖}
  • E1√2−1√2𝑖,−1√2+1√2𝑖

Q2:

Given that 𝑧=−28+96𝑖, determine the square roots of 𝑧 without first converting it to trigonometric form.

  • A−725+2425𝑖,−−725+2425𝑖
  • B(6+8𝑖),−(6+8𝑖)
  • C(8+6𝑖),−(8+6𝑖)
  • D2425−725𝑖,−2425−725𝑖
  • E√2−12𝑖,−√2−12𝑖

Q3:

Use De Moivre’s theorem to find the two square roots of 165𝜋3−𝑖5𝜋3cossin.

  • A±4𝑖
  • B±12−1√2𝑖
  • C±2√3+2𝑖
  • D±√32+12𝑖

Q4:

Given that 𝑧=𝜋6+𝑖𝜋6sincos, determine the cubic roots of 𝑧.

  • A5𝜋9+𝑖5𝜋9cossin, −7𝜋9+𝑖−7𝜋9cossin, −𝜋9+𝑖−𝜋9cossin
  • B7𝜋9+𝑖7𝜋9cossin, −5𝜋9+𝑖−5𝜋9cossin, 𝜋9+𝑖𝜋9cossin
  • C5𝜋18+𝑖5𝜋18cossin, 17𝜋18+𝑖17𝜋18cossin, −7𝜋18+𝑖−7𝜋18cossin
  • D7𝜋9+𝑖7𝜋9cossin, −8𝜋9+𝑖−8𝜋9cossin, −5𝜋9+𝑖−5𝜋9cossin
  • E−7𝜋9+𝑖−7𝜋9cossin, −4𝜋9+𝑖−4𝜋9cossin, −𝜋9+𝑖−𝜋9cossin

Q5:

Find the possible real values of 1√3𝑖+𝑖.

  • A−2, 0, and 2
  • B−1, 0, and 1
  • C−2√3, 0, and 2√3
  • D−1√3, 0, and 1√3
  • E−13, 0, and 13

Q6:

Solve 𝑧=16√2+16𝑖√2.

  • A𝑧=2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒
  • B𝑧=2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒
  • C𝑧=4𝑒, 4𝑒, 4𝑒, 4𝑒, 4𝑒
  • D𝑧=4𝑒, 4𝑒, 4𝑒, 4𝑒, 4𝑒
  • E𝑧=32𝑒, 32𝑒, 32𝑒, 32𝑒, 32𝑒

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 𝑧=125𝑒. What are their geometrical properties?

  • A5𝑒, 5𝑒, 5𝑒, 5𝑒, 5𝑒, 5𝑒; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius 5.
  • B√5𝑒, √5𝑒, √5𝑒, √5𝑒, √5𝑒, √5𝑒; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius √5.
  • C5𝑒, 5𝑒, 5𝑒, 5𝑒, 5𝑒, 5𝑒; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius √5.
  • D√5𝑒, √5𝑒, √5𝑒, √5𝑒, √5𝑒, √5𝑒; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius √5.
  • E125𝑒, 125𝑒, 125𝑒, 125𝑒, 125𝑒, 125𝑒; the roots lie at a straight line that passes through the origin.

State the 6th roots of unity.

  • A𝑒, 𝑖, 𝑒, 𝑒, −𝑖, 𝑒
  • B1, 𝑒, 𝑖, −1, 𝑒, −𝑖
  • C1, 𝑒, 𝑒, 𝑒, 𝑒, 𝑒
  • D1, 𝑒, 𝑒, −1, 𝑒, 𝑒
  • E1, 𝑒, 𝑒, −1, 𝑒, 𝑒

What is the relationship between the 6th roots of unity and the solutions to the equation 𝑧=125𝑒?

  • AThe solutions to the equation are the 6th roots of unity multiplied by 5𝑒.
  • BThe solutions to the equation are the 6th roots of unity multiplied by 125𝑒.
  • CThe solutions to the equation are the 6th roots of unity multiplied by 5𝑒.
  • DThe solutions to the equation are the 6th roots of unity multiplied by √5𝑒.
  • EThe solutions to the equation are the 6th roots of unity multiplied by √5𝑒.

Q8:

Find the roots of 𝑧+16=0.

  • A√2𝑒, √2𝑒, √2𝑒, √2𝑒, √2𝑒, √2𝑒, √2𝑒, √2𝑒.
  • B√2𝑒, √2𝑒, √2𝑒, √2𝑒, √2𝑒, √2𝑒, √2𝑒, √2𝑒
  • C2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒
  • D𝑒, 𝑒, 𝑒, 𝑒, 𝑒, 𝑒, 𝑒, 𝑒
  • E2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒, 2𝑒

The complex numbers representing the roots of 𝑧+16=0 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 (−1,2) with one vertex at the origin.

Give your answer as exact Cartesian coordinates.

  • A(0,0), 3+2√32,√3−62, 1+2√32,√3−22, (2,−4), 1−2√32,−√3−22, 3−2√32,−√3−62
  • B(0,0), −2−√32,−2√3+12, −6−√32,−2√3+12, (−4,2), −6+√32,2√3+32, −2+√32,2√3+12
  • C(0,0), −1+2√32,√3+22, −3+2√32,√3+62, (−2,4), −3−2√32,−√3+62, −1−2√32,−√3+22
  • D(0,0), (−2,0), (−3,2), (−2,4), (0,4), (1,2)
  • E(0,0), 2+√32,2√3−12, 6+√32,2√3−32, (4,−2), 6−√32,−2√3−32, 2−√32,−2√3−12

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), 3√213𝜋20,3√213𝜋20sincos, 3√2−19𝜋20,3√2−19𝜋20sincos, 3√2−11𝜋20,3√2−11𝜋20sincos, 3√2−3𝜋20,3√2−3𝜋20sincos
  • B(3,3), 3√213𝜋20,3√213𝜋20cossin, 3√2−19𝜋20,3√2−19𝜋20cossin, 3√2−11𝜋20,3√2−11𝜋20cossin, 3√2−3𝜋20,3√2−3𝜋20cossin
  • C(3,3), 3√211𝜋15,3√211𝜋15cossin, 3√2−13𝜋15,3√2−13𝜋15cossin, 3√2−7𝜋15,3√2−7𝜋15cossin, 3√2−𝜋15,3√2−𝜋15cossin
  • D(3,3), 313𝜋20,313𝜋20sincos, 3−19𝜋20,3−19𝜋20sincos, 3−11𝜋20,3−11𝜋20sincos, 3−3𝜋20,3−3𝜋20sincos
  • E(3,3), 313𝜋20,313𝜋20cossin, 3−19𝜋20,3−19𝜋20cossin, 3−11𝜋20,3−11𝜋20cossin, 3−3𝜋20,3−3𝜋20cossin

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