SOLUTION: <pre>Write in simplified radical form by rationalizing the denominator. _ <font face = "symbol">Ö</font>3 - 2 ————<u>—</u>————— -5<font face = "symbol">Ö</font>3 + 8</p

To rationalize the denominator means to get rid of any
radicals in the denominator.

   _
  Ö3 - 2
——————————
 -5Ö3 + 8

Since the denominator contains more than one term, at 
least one of which is a square root radical.


We create the "conjugate surd" of the denominator by 
changing only the sign of one of the (in this 'the only')
square root radical term. This expression we create is 
   _
+5Ö3 + 8

Then we place it over itself:

    _
  5Ö3 + 8
 ————————
  5Ö3 + 8

which equals 1.  Therefore we can multiple the original
expression by this fraction which does not change its 
numerical value, because we can multiply any expression
by 1 without changing its numerical value.

   _           _
  Ö3 - 2     5Ö3 + 8
————————— × —————————
 -5Ö3 + 8    5Ö3 + 8

Indicate the product of the numerators and denominators 
as one fraction:
   _           _
  Ö3 - 2     5Ö3 + 8
————————— × —————————
 -5Ö3 + 8    5Ö3 + 8

   _        _
 (Ö3 - 2)(5Ö3 + 8)
———————————————————
(-5Ö3 + 8)(5Ö3 + 8)


Use FOIL to multiply out the numerator and denominator:

    _    _      _      _
  (Ö3)(5Ö3) + 8Ö3 - 10Ö3 - 16
——————————————————————————————
(-5Ö3)(5Ö3)  - 40Ö3 + 40Ö3 + 64
           _
   5·3 - 2Ö3 - 16
 —————————————————
    -25·3 + 64
          _
   15 - 2Ö3 - 16
 —————————————————
    -75 + 64
         _
  -1 - 2Ö3 
 ——————————
    -11

to get rid of the negative signs multiply by the
fraction (-1)/(-1)
               _
 -1    (-1 - 2Ö3)   
———— × —————————— 
 -1        -11 
           _
 -1(-1 - 2Ö3)   
—————————————— 
   (-1)(-11)

        _
  1 + 2Ö3
 ————————— 
    11

Edwin
AnlytcPhil@aol.com