The value of \(3+\frac{1}{\sqrt3}+\frac{1}{3+\sqrt3}+\frac{1}{\sqrt3-3}\) is

1/(3+sqrt3) + 1/(sqrt3-3) = [(sqrt3-3)+(sqrt3+3)]/(3-9) = 2sqrt3/(-6) = -1/sqrt3, which cancels the +1/sqrt3. Value = 3.