44444..4 = 4*(10^2018-1)/9

111...1 = (10^1010-1)/9

666...6 = 6(10^1009-1)/9

10=x

4*x^2018+x^1010-6*x^1009+1 = x^1009(4x^1009+x-6)+1 = x^1009(4x^1009+4)+1=4x^2018+4x^1009+1=(2x^1009+1)^2 = (2*10^1009+1)^2

Значит корень из числа равен 2*10^1009+1