Question 172899
{{{((x+y)/(x-y))/(1/x-1/y)}}} Start with the given expression



Let's focus on the denominator {{{1/x-1/y}}} for now




{{{1/x-1/y}}} Start with the denominator 



{{{y/(xy)-1/y}}} Multiply the first fraction by {{{y/y}}}



{{{y/(xy)-x/(xy)}}} Multiply the second fraction by {{{x/x}}}



{{{(y-x)/(xy)}}} Combine the fractions (this only possible now that the denominators are equal)



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So the expression 


{{{((x+y)/(x-y))/(1/x-1/y)}}}



now becomes



{{{((x+y)/(x-y))/((y-x)/(xy))}}}




{{{((x+y)/(x-y))((xy)/(y-x))}}} Multiply the first fraction {{{(x+y)/(x-y)}}} by the reciprocal of the second fraction {{{(y-x)/(xy)}}}



{{{((x+y)/(x-y))((xy)/(-(x-y)))}}} Factor {{{y-x}}} to get {{{-(x-y)}}}



{{{((x+y)/(x-y))(-(xy)/((x-y)))}}} Reduce



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



{{{-(xy(x+y))/((x-y)^2)}}} Simplify





So {{{((x+y)/(x-y))/(1/x-1/y)}}} simplifies to {{{-(xy(x+y))/((x-y)^2)}}}