If x = 2 - √3, find the value of (x - 1/x)³

X is defined as "2 - √3."  Subst. this value into the expression (x - 1/x)³.

                               1
We get (2 - √3 -
                            2 - √3

Concentrate on simplifying this expression before worrying about cubing it.

The LCD here is 2 - √3.  Multiply 2 - √3 by 2 - √3 and then divide the resulting product by 2 - √3 to obtain 2 fractions with the same denominators:

 (2 - √3)(2 - √3)             1
-    This works out to 4-4√3 + 3, or 1-4√3, less 1, 
     2 - √3                  2 - √3     over  the denominator   2 - √3 :

 -4√3

2 - √3

At this point you have 2 choices.  You could leave this fraction as is and then cube the whole fraction.  Or you could "rationalize the denominator" first and then cube the resulting fraction.  Your choice!  If you share your results I'd be happy to comment on them.