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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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