Cube roots of unity - Practice Questions & MCQ

Let z be the cube root of unity (1)

So, z^₃ = 1  

⇒ z^₃ - 1 = 0

⇒ (z - 1)(z^₂ + z + 1) = 0

⇒ z - 1 = 0  or z^₂ + z + 1 = 0

If the second root is represented by ⍵, then the third root will be represented by ⍵^₂ (we can check that by squaring the second root, we get the third root)

So, 1, ⍵, ⍵² are cube roots of unity and ⍵, ⍵² are the non-real complex root of unity.

Properties of Cube roots of unity

i) 1 + ⍵ + ⍵² = 0 and ⍵³ = 1 (Using sum and produc of roots relations for the equation z_³ - 1 = 0)

ii) To find ⍵ⁿ , first we write ⍵ in multiple of 3 with remainder being 0 or 1 or 2.

    Now ⍵ⁿ = ⍵^{3q + r} = (⍵³)^q·⍵^r  = ⍵^r (Where r is from 0,1,2)

    Eg, ⍵¹²¹ = ⍵^{3.40 + 1} = (⍵³)⁴⁰·⍵¹  = ⍵

iii) |⍵| = |⍵² | = 1, arg(⍵) = 2π/3, arg(⍵²) = 4π/3 or -2π/3

iv) We can see that ⍵ and ⍵²  differ by the minus sign of imaginary part hence 

v) Cube roots of -1 are -1, -⍵, -⍵²

vi) The cube roots of unity when represented on the complex plane has its point on vertices of triangle circumscribed by a unit circle whose one vertices lies on the     +ve X-axis.