Ok, I'll try to apply that technique to the expression:
cbrt(54+sqrt(2700))+cbrt(54sqrt(2700))
a = cbrt(54+sqrt(2700))
b = cbrt(54sqrt(2700))
a^3 + b^3 = 108
a*b = cbrt((54+sqrt(2700))*(54sqrt(2700))) = cbrt(216) = 6
x (the number we're trying to determine) = a + b
x^3 = (a+b)^3 = a^3 + 3(a^2)b + 3a(b^2) + b^3 = a^3 + b^3 + 3ab(a+b) = a^3 + b^3 + 3abx
x^3 = 108 + 18x
x^3  18x + 108 = 0 (which is exactly the cubic where I got my expression from). This is basically just the reverse of the depressed cubic formula.
