Expanding the given equation, we get:

(x-3)^4 + (x-6)^4 = (x-3)(x-3)(x-3)(x-3) + (x-6)(x-6)(x-6)(x-6)
= (x^2 - 6x + 9)(x^2 - 6x + 9) + (x^2 - 12x + 36)(x^2 - 12x + 36)
= (x^4 - 12x^3 + 45x^2 - 54x + 81) + (x^4 - 24x^3 + 216x^2 - 864x + 1296)
= 2x^4 - 36x^3 + 261x^2 - 918x + 1377
= 16

So the solution to the equation is x = 3.