The following identity shows the relationship between the sum and the sums of powers:
6*(x^4+y^4+z^4) = (x+y+z)^4 + 3*(x^2+y^2+z^2)^2 - 6*(x^2+y^2+z^2)(x+y+z)^2 + 8*(x^3+y^3+z^3)(x+y+z)

Let x + y + z = N and substitute:
6*11 = N^4 + 3*3^2 - 6*3*N^2 + 8*7*N
0 = N^4 - 18*N^2 + 56*N - 39
N=1, -5.4414, 2.2207 +/- 1.4953*i

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