Solve this:

2/(2-1/sqrt(2))

Possible answers:

a. (4+sqrt(2))/7
b. (4*sqrt(2)+2)/7
c. (2*sqrt(2)+1)/7
d. (8+2*sqrt(2))/7

    2
——————————
      1
2 - —————
      Ö2

First of all rationalize the denominator of

   1
 —————
   Ö2

by multiplying top and bottom by Ö2
           _       _
   1      Ö2      Ö2 
 ————— × ————— = —————
   Ö2     Ö2       2

And substitute this for the bottom right

    2
——————————
     Ö2
2 - ————
      2

Now we get a common denominator in the bottom

      2
———————————————
   4     Ö2
  ——— - —————
   2      2


      2
———————————————
    4 - Ö2
  ——————————
      2

Now write this as a division

            _ 
       4 - Ö2
2 ÷  ——————————
         2

Write the 2 over 1:

              _
 2       4 - Ö2
——— ÷  ——————————
 1         2

Invert the second fraction and multiply:

 2        2  
——— ×  ————————
 1      4 - Ö2


     4  
———————————
  4 - Ö2

Rationalize the denominator by multiplying
both numerator and denominator by 4 + Ö2
                 _
    4       4 + Ö2
———————— × ————————
 4 - Ö2     4 + Ö2
           _
    4(4 + Ö2)
——————————————————
 (4 - Ö2)(4 + Ö2)

Use FOIL on the bottom:
           _
    4(4 + Ö2)
———————————————————
16 + 4Ö2 - 4Ö2 - 2
           _
    4(4 + Ö2)
  ————————————
       14

Divide both the 4 and the 14 by 2

    2      _
    4(4 + Ö2)
  ————————————
       14
       7
          _
   2(4 + Ö2)
  ——————————
      7

Distribute out the top
         _ 
   8 + 2Ö2)
  ——————————
      7

That's choice (d).

Edwin
AnlytcPhil@aol.com