Cube roots of unity: 1, w, w^2 where w = (-1+isqrt(3))/2 = e^(2pi*i/3).

Fundamental Properties:

• w^3 = 1 and 1 + w + w^2 = 0
• w^2 = w-bar (they are conjugates)
• |w| = |w^2| = 1
• w^(-1) = w^2

Simplification Substitutions:

• 1 + w = -w^2 (most used substitution)
• 1 + w^2 = -w
• w + w^2 = -1

Useful Products:

• (1-w)(1-w^2) = 3
• (1+w)(1+w^2) = 1
• (a+bw+cw^2)(a+bw^2+cw) = a^2+b^2+c^2-ab-bc-ca

Factorizations:

• x^2+x+1 = (x-w)(x-w^2)
• x^3-1 = (x-1)(x-w)(x-w^2)
• a^3+b^3+c^3-3abc = (a+b+c)(a+bw+cw^2)(a+bw^2+cw)

Power Reduction: For any integer n, w^n = w^(n mod 3). This converts any power of w to one of {1, w, w^2}.

JEE Problem Strategy: When you see w in a problem:

1. Immediately write 1+w+w^2 = 0
2. Reduce all powers mod 3
3. Replace 1+w with -w^2 (or similar)
4. Simplify using w^3 = 1

These identities appear in determinants, binomial coefficients, and algebraic expression problems. Master the substitutions to solve in 30-60 seconds.