SOLUTION: Simplify 6 times the square root of 7/5 - 2 times the square root of 5/7 - 3 times the square root of 140

Simplify: 
  ___     ___     ___
6Ö7/5 - 2Ö5/7 - 3Ö140

We have to simplify each of the three radicals:

First term has this radical
      _____
     / 7
 \  / ---
  \/   5

We must rationalize the denominator by multiplying
under the radical by this fraction which equals 1:

 5
---
 5

      ________
     / 7   5
 \  / ---×---    = 
  \/   5   5

      _____        __
     / 35         Ö35      __
 \  / ----    =  ------ = Ö35/5 
  \/   5²           5


So now we have
  __       ___     ___
6Ö35/5 - 2Ö5/7 - 3Ö140

The second term has this radical
      _____
     / 5
 \  / ---
  \/   7

We must rationalize the denominator by multiplying
under the radical by this fraction which equals 1:

 7
---
 7

      ________
     / 5   7
 \  / ---×---    = 
  \/   7   7

      _____        __
     / 35         Ö35      __
 \  / ----    =  ------ = Ö35/7 
  \/   7²           7


So now we have
  __       __       ___
6Ö35/5 - 2Ö35/7 - 3Ö140
                         ___
The third term contains Ö140, which
we simplify by writing 140 as 4×35
 ___    ____    _  __      __     __
Ö140 = Ö4×35 = Ö4×Ö35 = 2×Ö35 = 2Ö35

So now we have 

  __       __         __
6Ö35/5 - 2Ö35/7 - 3·2Ö35
              
or

  __       __       __
6Ö35/5 - 2Ö35/7 - 6Ö35
                                        __
Now we factor out the common factor of Ö35
 __
Ö35(6/5 - 2/7 - 6)

Then we get a common denominator, LCD = 35,
inside the parentheses, and write

6/5 as 42/35
2/7 as 10/35
6 as 210/35
 __
Ö35(42/35 - 10/35 - 210/35) =
 __
Ö35(-178/35) =
     __
-178Ö35/35 

Edwin