# Worksheet: Arbitrary Roots of Complex Numbers

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

**Q2: **

Find the fourth roots of , giving your answers in trigonometric form.

- A, , ,
- B, , ,
- C, , ,
- D, , ,

**Q3: **

Determine the solution set of the equation in , expressing the solutions in exponential form.

- A
- B
- C
- D

**Q4: **

Determine the solution set of the equation in .

- A
- B
- C
- D

**Q5: **

Determine the square roots of , given that .

- A
- B
- C
- D
- E

**Q6: **

Given that , determine the square roots of without first converting it to trigonometric form.

- A
- B
- C
- D
- E

**Q7: **

Given that , determine the square roots of without first converting it to trigonometric form.

- A
- B
- C
- D
- E

**Q8: **

Without first converting to trigonometric form, determine the square roots of , where .

- A,
- B,
- C1,
- D,

**Q9: **

Determine the two square roots of without first converting it to trigonometric form.

- A
- B
- C
- D
- E

**Q10: **

Find the square roots of , giving your answers in trigonometric form.

- A,
- B,
- C,
- D,
- E,

**Q11: **

Determine, in trigonometric form, the square roots of .

- A,
- B,
- C,
- D,
- E,

**Q12: **

Use De Moivre’s theorem to find the two square roots of .

- A
- B
- C
- D

**Q13: **

Given that , find the two square roots of without first expressing in trigonometric form.

- A
- B
- C
- D
- E

**Q14: **

Given that , find .

- A
- B
- C
- D
- E

**Q15: **

Given that , find all possible real values of and .

- A
- B
- C
- D
- E

**Q16: **

Given that , determine the cubic roots of .

- A, ,
- B, ,
- C, ,
- D, ,
- E, ,

**Q17: **

Given that , find the square roots of in trigonometric form.

- A,
- B,
- C,
- D,
- E,

**Q18: **

Find the two square roots of in , without first expressing in trigonometric form.

- A
- B
- C
- D

**Q19: **

Find the possible values of .

- A, 0, and 2
- B, 0, and 1
- C, 0, and
- D, 0, and
- E, 0, and

**Q20: **

Find the solutions to the equation . What are their geometrical properties?

- A, , , , , ; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius 5.
- B, , , , , ; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius .
- C, , , , , ; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius .
- D, , , , , ; the roots lie at the vertices of a regular hexagon centered at the origin, inscribed in a circle of radius .
- E, , , , , ; the roots lie at a straight line that passes through the origin.

State the 6th roots of unity.

- A, , , , ,
- B1, , , , ,
- C1, , , , ,
- D1, , , , ,
- E1, , , , ,

What is the relationship between the 6th roots of unity and the solutions to the equation ?

- AThe solutions to the equation are the 6th roots of unity multiplied by .
- BThe solutions to the equation are the 6th roots of unity multiplied by .
- CThe solutions to the equation are the 6th roots of unity multiplied by .
- DThe solutions to the equation are the 6th roots of unity multiplied by .
- EThe solutions to the equation are the 6th roots of unity multiplied by .

**Q21: **

Find the roots of .

- A, , , , , , , .
- B, , , , , , ,
- C, , , , , , ,
- D, , , , , , ,
- E, , , , , , ,

The complex numbers representing the roots of are each squared to form the vertices of a new shape. What is the area of this shape?

**Q22: **

Find the coordinates of the vertices of a regular hexagon centered at with one vertex at the origin.

Give your answer as exact Cartesian coordinates.

- A, , , , ,
- B, , , , ,
- C, , , , ,
- D, , , , ,
- E, , , , ,

**Q23: **

Solve .

- A, , , ,
- B, , , ,
- C, , , ,
- D, , , ,
- E, , , ,

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.

**Q24: **

A small robot is programmed to travel units forward and then turn to the left by an angle of . If it does this times, how far will it be from the starting point? Give you answer in exact form.

- A
- B
- C
- D
- E

**Q25: **

Find the coordinates of the vertices of a regular pentagon centered at the origin with one vertex at .

Give your answer as exact Cartesian coordinates.

- A, , , ,
- B, , , ,
- C, , , ,
- D, , , ,
- E, , , ,