We can simplify the given expression by rationalizing the denominator of x:
x = (root 3 + root 2) / (root 3 - root 2)
Multiplying the numerator and denominator by the conjugate of the denominator, we get:
x = [(root 3 + root 2) * (root 3 + root 2)] / [(root 3 - root 2) * (root 3 + root 2)]
x = (3 + 2root 6 + 2) / (3 - 2)
x = 5 + 2root 6
Now, we can substitute this value of x in the given expression:
x^4 + 1 / x^4
= (5 + 2root 6)^4 + 1 / (5 + 2root 6)^4
= 50625 + 36000root 6 + 9600 + 1280root 6 / 50625 - 36000root 6 + 9600 + 1280root 6
= (50625 + 9600) + (36000root 6 + 1280root 6) / (50625 + 9600) - (36000root 6 - 1280root 6)
= 60225 + 37280root 6 / 41025 - 34720root 6
= (60225 + 37280root 6) * (41025 + 34720root 6) / (41025 - 34720root 6) * (41025 + 34720root 6)
= 2470745625 / 182739225
= 13.509
Therefore, x^4 + 1 / x^4 is approximately equal to 13.509.