# Worksheet: Cube Roots of Unity

In this worksheet, we will practice identifying the cubic roots of unity using de Moivre’s theorem.

Q1:

If , where is a positive integer and is one of the complex cubic roots of unity, then what is ?

Q2:

Write in its simplest form.

• A
• B
• C1

Q3:

Write in its simplest form.

• A
• B
• C1

Q4:

Evaluate , where is a primitive cubic root of unity.

• A19,683
• B
• C
• D
• E

Q5:

Evaluate , where is a primitive cubic root of unity.

• A
• B
• C
• D1
• E

Q6:

Evaluate where is a complex cube root of unity.

Q7:

Evaluate where is a complex cube root of unity.

• A
• B
• C1
• D0

Q8:

Evaluate where is a complex cube root of unity.

• A
• B
• C0
• D

Q9:

What is ?

• A49
• B48
• C0
• D57

Q10:

Evaluate where is a complex cube root of unity.

Q11:

If , where and are real numbers and is a complex cube root of unity, then what is ?

• A
• B
• C
• D

Q12:

Evaluate .

• A1
• B
• C
• D
• E

Q13:

What is the smallest positive integer value of for which where is a complex cube root of unity?

Q14:

Evaluate , where is a complex cube root of unity.

• A
• B
• C
• D
• E

Q15:

Evaluate .

• A
• B
• C1
• D
• E

Q16:

Given that is one of the complex cubic roots of unity, find the complex conjugate of .

• A
• B
• C
• D

Q17:

Given that is one of the complex cubic roots of unity, find the complex conjugate of .

• A
• B1
• C
• D

Q18:

Evaluate , where is a primitive cubic root of unity.

• A
• B1
• C
• D
• E

Q19:

Evaluate where is a complex cube root of unity.

Q20:

Evaluate , where is a primitive cubic root of unity.

• A
• B
• C
• D1
• E

Q21:

Write in its simplest form.

• A
• B1
• C

Q22:

Write in its simplest form.

• A
• B1
• C

Q23:

Write in its simplest form.

• A
• B1
• C

Q24:

Write in its simplest form.

• A1
• B
• C

Q25:

Write in its simplest form.

• A
• B1
• C