x^4+y^4+z^4= 0.5 is the answer.
Explanation:
we know that given,
x+y+z=0
x^2+y^2+z^2=1
x^4+y^4+z^4=?
∴ 2 (xy+yz+zx) = (x+y+z)^2 - (x^2+y^2+z^2) = -1
(xy + yz+ zx) = -1 ÷ 2
We also know the formula,
6xyz = (x+y+z)^3- 3 (x+y+z) (x^2+y^2+z^2) + 2(x^3+y^3+z^3) = 1
xyz = 1 ÷ 6
x^4+y^4+z^4 = (x^2+y^2+z^2)^2 - 2(x^2y^2+y^2z^2+z^2x^2)
= (x^2+y^2+z^2)^2 - 2 ((xy+yz+zx)^2 - 2xyz (x+y+z))
= 1 - 2( 1÷4 ) - 2( 1÷6 ) (0)
= 0.5