also, this is not right. we can distribute powers across products, not sums
$\displaystyle (a + b)^2 \ne a^2 + b^2$ for instance, but rather $\displaystyle (a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2$
similarly, $\displaystyle (x^{-1} + y^{-1})^{-2} = \frac 1{(x^{-1} + y^{-1})^2} = \frac 1{x^{-2} + 2(xy)^{-1} + y^{-2}}$
if you don't like negative powers in the denominator, you could approach it this way
$\displaystyle (x^{-1} + y^{-1})^{-2} = \left( \frac 1x + \frac 1y \right)^{-2} = \left( \frac {x + y}{xy} \right)^{-2} = \frac {x^2y^2}{(x + y)^2} = \frac {x^2y^2}{x^2 + 2xy + y^2}$
also, please post new problems in a new thread. it saves me the work of having to split the thread up