Question 131897
{{{(sqrt(x)-sqrt(y))/(sqrt(x)+sqrt(y))}}} Start with the given expression



{{{((sqrt(x)-sqrt(y))/(sqrt(x)+sqrt(y)))((sqrt(x)-sqrt(y))/(sqrt(x)-sqrt(y)))}}} Multiply by the fraction by {{{(sqrt(x)-sqrt(y))/(sqrt(x)-sqrt(y))}}}. Note  {{{sqrt(x)-sqrt(y)}}} is the conjugate of {{{sqrt(x)+sqrt(y)}}}.



{{{((sqrt(x)-sqrt(y))(sqrt(x)-sqrt(y)))/((sqrt(x)+sqrt(y))(sqrt(x)-sqrt(y)))}}} Combine the fractions



{{{((sqrt(x)-sqrt(y))(sqrt(x)-sqrt(y)))/(sqrt(x)*sqrt(x)+sqrt(x)*sqrt(y)-sqrt(y)*sqrt(x)+sqrt(y)*sqrt(y))}}} Foil the denominator



{{{((sqrt(x)-sqrt(y))(sqrt(x)-sqrt(y)))/(sqrt(x)*sqrt(x)+sqrt(y)*sqrt(y))}}} Cancel like terms



{{{((sqrt(x)-sqrt(y))(sqrt(x)-sqrt(y)))/(x-y)}}} Multiply



{{{(sqrt(x)sqrt(x)-2sqrt(x)sqrt(y)+sqrt(y)sqrt(y))/(x-y)}}} Foil the numerator



{{{(x-2sqrt(x)sqrt(y)+y)/(x-y)}}} Multiply




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Answer:

So {{{(sqrt(x)-sqrt(y))/(sqrt(x)+sqrt(y))}}} simplifies to {{{(x-2sqrt(x)sqrt(y)+y)/(x-y)}}}