# How can I simplify this expression

I know that I cannot just do $$\frac{x^{-1}}{x^{-3}}$$ and $$\frac{y^{-1}}{y^{-3}}$$ and get $$x^{2}+y^{2}$$ but how can i factor either the numerator or denominator to get something to cancel out?

$$\dfrac{x^{-1}+y^{-1}}{x^{-3}+y^{-3}}=\dfrac{x^3y^3(x^{-1}+y^{-1})}{x^3y^3(x^{-3}+y^{-3})}= \dfrac{x^{2}y^3+x^3y^{2}}{x^{3}+y^{3}}=\dfrac{(x^2y^2)(x+y)}{x^{3}+y^{3}}=\dfrac{(x^2y^2)(x+y)}{(x+y)(x^2-xy+y^2)}=\dfrac{(x^2y^2)}{(x^2-xy+y^2)}$$
It is $$\frac{\frac{x+y}{xy}}{\frac{x^3+y^3}{x^3y^3}}=\frac{(x+y)x^3y^3}{xy(x^3+y^3)}$$ and now you can use (after cancelling $$xy$$) that $$x^3+y^3=(x+y)(x^2-xy+y^2)$$
$$\frac1{x^{-2}-x^{-1}y^{-1}+y^{-2}},$$ by factorization.