If α, β, γ are cube roots of −8, then what is (α²p² + β²q² + γ²r²)/(β²p² + γ²q² + α²r²) equal to?

The cube roots of −8 are −2, −2ω, −2ω², where ω is a primitive cube root of unity. Squaring gives α² = 4, β² = 4ω², γ² = 4ω (taking α = −2, β = −2ω, γ = −2ω²). Then numerator = 4(p² + ω²q² + ωr²) and denominator = 4(ω²p² + ωq² + r²). Factoring ω² from the denominator: ω²(p² + ω²·q²/ω² ... ). Multiplying numerator and denominator appropriately and using ω³ = 1, the ratio simplifies to γ/α.