X+y+z=0

x^2+y^2+z^2=1

x^4+y^4+z^4=?

x^4+y^4+z^4 = 0.5 is the answer.

Step-by-step 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