Question 601808
I think the question is to simplify 

{{{(X^(-2)-Y^(-2))/(X^(-1)+Y^(-1))}}}

Now using your first step. we get
{{{((1/(X^2))-(1/(Y^2)))/((1/X)+(1/Y))}}}
Taking the LCM of Numerator & Denominator separately, we get
{{{((Y^2-X^2)/(X^2*Y^2))/((Y+X)/(X*Y))}}}
i.e.
{{{((Y^2-X^2)*X*Y)/(X^2*Y^2*(Y+X))}}}
i.e.
{{{((Y-X)(Y+X)*X*Y)/(X^2*Y^2*(Y+X))}}}  [Using identity {{{(a^2-b^2)=(a+b)(a-b)}}}]

i.e.{{{(Y-X)/(X*Y)}}}

This is the required solution, but It can further be written as

{{{Y/(X*Y)-X/(X*Y)}}}
i.e.
{{{1/X-1/Y}}}
OR
{{{X^(-1)-Y^(-1)}}}