**Solution:**

To find the value of x, we have the equation:

x = 5 - 2√6

To simplify this expression, we need to rationalize the denominator of the radical term. The conjugate of 2√6 is -2√6, so we can multiply both the numerator and denominator by -2√6:

x = (5 - 2√6) * (-2√6) / (-2√6)

Expanding and simplifying, we get:

x = (-10√6 + 4 * 6) / (-12)

x = (-10√6 + 24) / (-12)

x = (24 - 10√6) / (-12)

x = -2 + (5/2)√6

So, x is equal to -2 + (5/2)√6.

Now, let's find x^3 + 1/x^3.

Substituting the value of x we found earlier, we have:

x^3 + 1/x^3 = (-2 + (5/2)√6)^3 + 1/(-2 + (5/2)√6)^3

Expanding and simplifying this expression may be quite complex, so let's use a shortcut.

We can use the formula for the sum of cubes:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Using this formula, we can rewrite x^3 as:

x^3 = (-2 + (5/2)√6)^3

x^3 = (-2)^3 + 3(-2)^2((5/2)√6) + 3(-2)((5/2)√6)^2 + ((5/2)√6)^3

Simplifying further, we have:

x^3 = -8 + 60√6 - 150 + (125/8)√6

x^3 = -158 + (173/8)√6

Similarly, we can find 1/x^3 as:

1/x^3 = 1/(-2 + (5/2)√6)^3

1/x^3 = 1/(-158 + (173/8)√6)

Now, we can evaluate the expression x^3 + 1/x^3:

x^3 + 1/x^3 = (-158 + (173/8)√6) + (1/(-158 + (173/8)√6))

Simplifying this expression may require further calculations, but this is the general process to find x^3 + 1/x^3 using the given value of x.