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In this lesson, we will learn how to identify the cubic roots of unity using de Moivre's theorem.

Q1:

Evaluate 𝜔 where 𝜔 is a complex cube root of unity.

Q2:

Evaluate 7 + 𝜔 + 𝜔 where 𝜔 is a complex cube root of unity.

Q3:

Evaluate 𝜔 + 𝜔 + 𝜔 + 𝜔 + ⋯ + 𝜔 where 𝜔 is a complex cube root of unity.

Q4:

If 1 + 𝜔 = 𝑎 + 𝑏 𝜔 2 1 1 , where 𝑎 and 𝑏 are real numbers, then what is ( 𝑎 , 𝑏 ) .

Q5:

Evaluate − 6 − 6 𝜔 − 6 𝜔 where 𝜔 is a complex cube root of unity.

Q6:

Evaluate 2 + 2 𝜔 + 2 𝜔 where 𝜔 is a complex cube root of unity.

Q7:

Evaluate 1 + 𝜔 2 − 1 3 3 .

Q8:

Given that 𝜔 is one of the complex cubic roots of unity, find the complex conjugate of 1 + 𝜔 .

Q9:

Evaluate 1 + 𝜔 , where 𝜔 is a primitive cubic root of unity.

Q10:

Evaluate 1 + 𝜔 , where 𝜔 is a primitive cubic root of unity.

Q11:

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

Q12:

If 𝑧 = 𝜔 𝑥 , where 𝑥 is a positive integer and 𝜔 is one of the complex cubic roots of unity, then what is | 𝑧 | ?

Q13:

What is the smallest positive integer value of 𝑛 for which 1 + 𝜔 = ( 1 + 𝜔 ) where 𝜔 is a complex cube root of unity?

Q14:

Write 𝜔 − 1 4 9 in its simplest form.

Q15:

Write 𝜔 − 4 in its simplest form.

Q16:

Write 𝜔 − 2 1 in its simplest form.

Q17:

Write 𝜔 − 8 5 in its simplest form.

Q18:

Write 𝜔 1 1 in its simplest form.

Q19:

Write 𝜔 9 in its simplest form.

Q20:

Write 𝜔 1 3 0 in its simplest form.

Q21:

Evaluate 1 + 𝜔 2 3 5 .

Q22:

Evaluate 1 + 𝜔 2 2 8 .

Q23:

Evaluate ( 1 − 𝜔 ) , where 𝜔 is a primitive cubic root of unity.

Q24:

Evaluate ( 1 − 𝜔 ) , where 𝜔 is a primitive cubic root of unity.

Q25:

What is 6 𝑟 = 0 𝑟 ( 8 + 𝜔 ) ?

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