ok, set sqrt(11+sqrt72)) + sqrt(11-sqrt(72)) = x
(by the way, sqrt(72) = 6*sqrt(2), which i'll just call 6r2 for simplicity)
square both sides, we get
(sqrt(11+sqrt72)) + sqrt(11-sqrt(72)))^2 = x^2
simplify and you get
x=6 (well, plus or minus, but we know it must be positive since the addition of two non-complex roots must be >= 0)
Yes, surd: an expression involving radicals. I just didn't want to have to type it out. Just square the expression, simplify and take the square root, et voila, we have 6, with the appropriate choice of sign.