# 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.

Q1:

Find the cube roots of 64, giving your answers in trigonometric form.

• A, ,
• B, ,
• C, ,
• D, ,

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.

• 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 .