Once we realize to multiply the second equation by 4, the terms in the equations look very much like the expansion of $ (x+y)^4 $, with some negative signs appearing. Closer inspection reveals that the two given equations are equivalent to the real and imaginary parts of a single statement involving complex numbers: \[ (x+iy)^4 = 8 + 8i\sqrt{3} = 16e^{\pi i/3}.\]In other words, we need only find the fourth root of $ 16e^{\pi i/3} $ in the first quadrant, where $ x $ and $ y $ will both be positive. Thus $ x+iy=2e^{\pi i/12} $; taking the real part yields our answer $ x=2\cos\left(\frac{\pi}{12}\right) = \boxed{2\cos 15^{\circ}} $.