# Lesson Worksheet: De Moivre’s Theorem Mathematics

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:

Use De Moivreโs theorem to find the two square roots of .

• A
• B
• C
• D
• E

Q2:

If , what is , where is an integer?

• A
• B
• C
• D

Q3:

What is ?

• A
• B
• C
• D
• E2

Q4:

Use De Moivreโs theorem to find the two square roots of .

• A
• B
• C
• D
• E

Q5:

Use De Moivreโs theorem to find the two square roots of .

• A
• B
• C
• D
• E

Q6:

What is ?

• A
• B
• C
• D16
• E2

Q7:

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

• A,
• B,
• C,
• D,

Q8:

Simplify .

Q9:

Consider the complex number .

Find the modulus of .

• A
• B
• C1
• D3
• E

Hence, find the modulus of .

• A10
• B
• C243
• D
• E

Q10:

Determine, in trigonometric form, the square roots of .

• A,
• B,
• C,
• D,

Q11:

• A
• B
• C
• D
• E

Q12:

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 .

Q13:

• A
• B
• C
• D
• E

Q14:

• A
• B
• C
• D

Q15:

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

• A
• B
• C
• D
• E

Q16:

Given that , find in exponential form.

• A
• B
• C
• D
• E

Q17:

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

• A, ,
• B, ,
• C, ,
• D27, ,
• E81, ,

Q18:

If , what is ?

• A
• B
• C
• D
• E

Q19:

• A
• B
• C
• D

Q20:

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

• A
• B
• C
• D
• E

Q21:

Given that and , where , express in trigonometric form.

• A
• B
• C
• D

Q22:

Given that and , determine the principal amplitude of .

• A
• B
• C
• D

Q23:

Given that , determine the principal amplitude of .

Q24:

Consider the equation .

Express in polar form using the general form of the argument.

• A for
• B for
• C for
• D for
• E for

By applying de Moivreโs theorem to the left-hand side, rewrite the equation in polar form.

• A
• B
• C
• D
• E

By equating the moduli and arguments and considering different values of the general argument, find the 3 cube roots of , expressing them in exponential form.

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Q25: