SOLUTION: Rationalize the denominator. (x^1/2-y^1/2)/(x^1/2+y^1/2)

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Question 392872: Rationalize the denominator.
(x^1/2-y^1/2)/(x^1/2+y^1/2)
Found 2 solutions by stanbon, MathLover1:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Rationalize the denominator.
(x^1/2-y^1/2)/(x^1/2+y^1/2)
------
[sqrt(x)-sqrt(y)]^2/{[sqrt(x)+sqrt(y)]}
----
= [sqrt(x)-sqrt(y)]^2/[x-y]
=======================
cheers,
Stan H.

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

%28x%5E%281%2F2%29-y%5E%281%2F2%29%29%2F%28x%5E%281%2F2%29%2By%5E%281%2F2%29%29


%28sqrt%28x%29+-+sqrt%28y%29%29%2F%28sqrt%28x%29+%2B+sqrt%28y%29%29

Multiply by the fraction by %28sqrt%28x%29-sqrt%28y%29%29%2F%28sqrt%28x%29-sqrt%28y%29%29. Note sqrt%28x%29-sqrt%28y%29 is the conjugate of sqrt%28x%29%2Bsqrt%28y%29.


Combine the fractions


Foil the denominator


Cancel like terms


%28%28sqrt%28x%29-sqrt%28y%29%29%28sqrt%28x%29-sqrt%28y%29%29%29%2F%28x-y%29 Multiply


%28sqrt%28x%29sqrt%28x%29-2sqrt%28x%29sqrt%28y%29%2Bsqrt%28y%29sqrt%28y%29%29%2F%28x-y%29 Foil the numerator


%28x-2sqrt%28x%29sqrt%28y%29%2By%29%2F%28x-y%29 Multiply

Answer:

So %28sqrt%28x%29-sqrt%28y%29%29%2F%28sqrt%28x%29%2Bsqrt%28y%29%29 simplifies to


%28x-2sqrt%28x%29sqrt%28y%29%2By%29%2F%28x-y%29

Rule : If you have two positive numbers x and y, then


+sqrt%28x%29%2Asqrt%28y%29=sqrt%28x%2Ay%29


%28x-2sqrt%28x%2Ay%29%2By%29%2F%28x-y%29