Question 65449
<pre><font size = 5><b>Simplify: 
  ___     ___     ___
6<font face = "symbol">Ö</font>7/5 - 2<font face = "symbol">Ö</font>5/7 - 3<font face = "symbol">Ö</font>140

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         <font face = "symbol">Ö</font>35      __
 \  / ----    =  ------ = <font face = "symbol">Ö</font>35/5 
  \/   5²           5


So now we have
  __       ___     ___
6<font face = "symbol">Ö</font>35/5 - 2<font face = "symbol">Ö</font>5/7 - 3<font face = "symbol">Ö</font>140

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         <font face = "symbol">Ö</font>35      __
 \  / ----    =  ------ = <font face = "symbol">Ö</font>35/7 
  \/   7²           7


So now we have
  __       __       ___
6<font face = "symbol">Ö</font>35/5 - 2<font face = "symbol">Ö</font>35/7 - 3<font face = "symbol">Ö</font>140
                         ___
The third term contains <font face = "symbol">Ö</font>140, which
we simplify by writing 140 as 4×35
 ___    ____    _  __      __     __
<font face = "symbol">Ö</font>140 = <font face = "symbol">Ö</font>4×35 = <font face = "symbol">Ö</font>4×<font face = "symbol">Ö</font>35 = 2×<font face = "symbol">Ö</font>35 = 2<font face = "symbol">Ö</font>35

So now we have 

  __       __         __
6<font face = "symbol">Ö</font>35/5 - 2<font face = "symbol">Ö</font>35/7 - 3·2<font face = "symbol">Ö</font>35
              
or

  __       __       __
6<font face = "symbol">Ö</font>35/5 - 2<font face = "symbol">Ö</font>35/7 - 6<font face = "symbol">Ö</font>35
                                        __
Now we factor out the common factor of <font face = "symbol">Ö</font>35
 __
<font face = "symbol">Ö</font>35(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
 __
<font face = "symbol">Ö</font>35(42/35 - 10/35 - 210/35) =
 __
<font face = "symbol">Ö</font>35(-178/35) =
     __
-178<font face = "symbol">Ö</font>35/35 

Edwin</pre>