Find x + y if the pair

We already know that x + y   = 3  .....  do you mean x  - y  ???

x^4 = y^4 + 18√3,  x^2 + y^2  =  6,  x + y  =  3

Using the first equation

x^4 - y^4  = 18√3      factor the left side

(x^2  - y^2) (x^2 + y^2)  =18√3        sub in the second equation

(x^2 - y^2) (6)  =  18√3      divide both sides  by 6

(x^2  - y^2)  = 3√3    factor again  

( x - y) (x + y)  = 3√3  sub in the third equation

(x - y)  (3)  =  3√3    divide both sides by 3

x  - y   =  √3