Strategy 1: Algebraic -> Polar for powers For $z^n$: convert z to polar, apply De Moivre. ALWAYS faster than repeated multiplication.

Strategy 2: z = x + iy for mixed z, z-bar equations When equation has both z and z-bar, substitute z = x+iy, separate real and imaginary parts.

Strategy 3: |z|^2 = z*z-bar interchange Seeing |z|^2? Replace with zz-bar. Seeing zz-bar? It's |z|^2 (real).

Strategy 4: Geometry for locus |z-z0| type => circle. Don't expand algebraically if you can identify the shape geometrically.

Strategy 5: 1+w+$w^2$ = 0 immediately For any cube root of unity problem. Then: 1+w = -$w^2$, 1+$w^2$ = -w, w+$w^2$ = -1.

Strategy 6: Polar form for quotients arg$\frac{z1}{z2}$ = arg(z1) - arg(z2). Don't compute the actual quotient if only argument is needed.

Strategy 7: Conjugate for division Multiply top and bottom by conjugate of denominator to rationalize.

Strategy 8: Check quadrant for argument After computing reference angle, ALWAYS verify which quadrant z is in. Q2 and Q3 are the most error-prone.

Time allocation: JEE complex number questions should take 2-3 minutes. If you're spending more, you're likely using the wrong representation.