Real numbers a, b and c satisfy a + b + c = 0 and a² + b² + c² = 6. Find the value of a⁴ + b⁴ + c⁴.

From (a + b + c)² = a² + b² + c² + 2(ab + bc + ca): 0 = 6 + 2(ab + bc + ca), so ab + bc + ca = −3.

(ab + bc + ca)² = a²b² + b²c² + c²a² + 2abc(a + b + c). Since a + b + c = 0, a²b² + b²c² + c²a² = 9.

a⁴ + b⁴ + c⁴ = (a² + b² + c²)² − 2(a²b² + b²c² + c²a²) = 36 − 18 = 18.

Check: a = 1, b = 1, c = −2 satisfies both conditions and gives 1 + 1 + 16 = 18 ✓.