I am using the notations a^b = a to the b power, and sqrt(c) = the positive square root of c.

Expanding out (x + (2/x))^3, we get x^3 + 6x + 12/x + 8/(x^3).

Rearranging the terms,

(x + (2/x))^3 = x^3 + 8/(x^3) + 6(x + 2/x)

Now, we are given (x + 2/x)^2 = 6. Therefore (x + 2/x) = sqrt(6) or (x + 2/x) = - sqrt(6).

so, we have (sqrt(6))^3 = x^3 + 8/(x^3) + 6*sqrt(6), or
................(-sqrt(6))^3 = x^3 + 8/(x^3) - 6*sqrt(6)

solving each: 6*sqrt(6) = x^3 + 8/(x^3) + 6*sqrt(6)
0 = x^3 + 8/(x^3)

-6*sqrt(6) = x^3 + 8/(x^3) - 6*sqrt(6)
0 = x^3 + 8/(x^3)

In either case, x^3 + 8/(x^3) = 0, which is what we were asked to find. Looks like irchans is correct, it all disappears into nothingness.