SOLUTION: <pre>For all x > 0 and y > 0, the radical expression _ <font face = "symbol">Ö</font>x ---<u>-</u>----<u>-</u>- 3<font face = "symbol">Ö</font>x -

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Question 29489: 
For all x > 0 and y > 0, the radical expression
          _
         Öx
     ----------
      3Öx - Öy

is equivalent to ____?
Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!
For all x > 0 and y > 0, the radical expression
sqrt      _
         Öx
     ----------
      3Öx - Öy

Form the conjugate of the denominator. The conjugate of a
binomial is a binomial whose first term is the same as the
first term of the given binomial and whose second term is the
same as the second term of the given binomial, except that
its sign is changed.  Therefore the conjugate of the
denominator 3Öx - Öy is the binomial 3Öx + Öy

Multiply the given expression by 1 written in the form of
(conjugate)/(conjugate) or (3Öx + Öy)/(3Öx + Öy)
          _          _    _
         Öx       (3Öx + Öy)
     ---------- · ----------
      3Öx - Öy    (3Öx + Öy)
	
or
           _   _    _ 
          Öx(3Öx + Öy)
     ----------------------
      (3Öx - Öy)(3Öx + Öy)
	

Use the distributive law on top, and FOIL on the bottom
                __    __
              3Öx² + Öxy 
     ----------------------------          
       9Öx² + 3Öxy - 3Öxy - Öy²

The middle two terms cancel on the bottom

                __    __
              3Öx² + Öxy 
     ----------------------------          
       9Öx² + 3Öxy - 3Öxy - Öy²

Take the square roots of the squares and get:
                     __
               3x + Öxy
              ----------
                9x - y              

Edwin
AnlytcPhil@aol.com
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