(a + b + c)² = 1

$$ \Rightarrow $$ a² + b² + c² + 2(ab + bc + ca) = 1

$$ \Rightarrow $$ a² + b² + c² = – 3 ….(i)

$$ \Rightarrow $$ ab + bc + ca = 2 ….(ii)

Squaring of equation (ii),

$$ \Rightarrow $$ a²b² + b²c² + c²a² + 2(ab²c + bc²a + ca²b) = 4

$$ \Rightarrow $$ a²b² + b²c² + c²a² + 2abc(a + b + c) = 4

$$ \Rightarrow $$ a²b² + b²c² + c²a² + 6 = 4

$$ \Rightarrow $$ a²b² + b²c² + c²a² = – 2 ….(iii)

Squaring of equation (i),

$$ \Rightarrow $$ a⁴ + b⁴ + c⁴ + 2(a²b² + b²c² + c²a²) = 9

$$ \Rightarrow $$ a⁴ + b⁴ + c⁴ – 4 = 9

$$ \Rightarrow $$ a⁴ + b⁴ + c⁴ = 13