# Worksheet: De Moivre’s Theorem

In this worksheet, we will practice finding powers and roots of complex numbers using De Moivre’s theorem to simplify calculations of powers and roots.

Q1:

What is ?

• A
• B
• C
• D
• E

Q2:

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

• A
• B
• C
• D
• E

Q3:

If , what is ?

• A
• B
• C
• D

Q4:

What is ?

• A
• B
• C
• D
• E2

Q5:

What is ?

• A
• B
• C
• D
• E

Q6:

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

• A
• B
• C
• D
• E

Q7:

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

• A
• B
• C
• D
• E

Q8:

What is ?

• A
• B
• C
• D16
• E2

Q9:

Given that , where and are real numbers, determine the value of and the value of .

• A,
• B,
• C,
• D,

Q10:

Simplify .

Q11:

Consider the complex number .

Find the modulus of .

• A
• B
• C1
• D3
• E

Hence, find the modulus of .

• A10
• B
• C243
• D
• E

Q12:

Determine, in trigonometric form, the square roots of .

• A,
• B,
• C,
• D,

Q13:

• A
• B
• C
• D
• E

Q14:

Consider the complex number .

Find the modulus of .

Find the argument of .

• A2
• B
• C
• D
• E

Hence, use the properties of multiplication of complex numbers in polar form to find the modulus and argument of .

• Amodulus = 8, argument =
• Bmodulus = , argument =
• Cmodulus = 8, argument =
• Dmodulus = , argument=
• Emodulus = , argument =

Hence, find the value of .

Q15:

• A
• B
• C
• D
• E

Q16:

• A
• B
• C
• D

Q17:

If , , and , what is the exponential form of ?

• A
• B
• C
• D
• E

Q18:

Given that , find in exponential form.

• A
• B
• C
• D
• E

Q19:

Find the possible values of , giving your answers in trigonometric form.

• A, ,
• B, ,
• C, ,
• D, ,

Q20:

If , what is ?

• A
• B
• C
• D
• E

Q21:

• A
• B
• C
• D

Q22:

Given that , , and , find , giving your answer in exponential form.

• A
• B
• C
• D
• E

Q23:

Given that and , where , express in trigonometric form.

• A
• B
• C
• D

Q24:

Given that and , determine the principal amplitude of .

• A
• B
• C
• D

Q25:

Given that , determine the principal amplitude of .