If α, β, γ are the cube roots of any number p(p < 0), then for any three numbers x, y, z;  is equal to:

Concept 

Cube Roots of unity are 1, ω and ω²

Here, ω =  and ω² = 

Property of cube roots of unity

• ω³ = 1
• 1 + ω + ω² = 0
• ω = 1 / ω ² and ω² = 1 / ω
• ω³ⁿ = 1

Calculation:

Given: α, β, γ are the cube roots

Let z³ = p (p < 0)

Let p  = - q where q > 0

⇒  z3 = -q 

So, the roots of the above equation will be, 

Assume α = q, β = qω, γ = qω²

(Because the magnitude of each root will be q)

Now,

            (∵ ω³ = 1)