$\frac{a^{-2} + 2a^{-1}b^{-1} + b^{-2}}{a^{-2} - b^{-2}} = \frac{(a^{-1} + b^{-1})^2}{(a^{-1} - b^{-1})(a^{-1} + b^{-1})} = \frac{a^{-1} + b^{-1}}{a^{-1} - b^{-1}} = \frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a} - \frac{1}{b}} = \frac{\frac{b + a}{ab}}{\frac{b - a}{ab}} = \frac{b + a}{ab} * \frac{ab}{b - a} = \frac{b + a}{b - a}$

$\frac{a^{-3} + b^{-3}}{a^{-2} - a^{-1}b^{-1} + b^{-2}} = \frac{(a^{-1} + b^{-1})(a^{-2} - a^{-1}b^{-1} + b^{-2})}{a^{-2} - a^{-1}b^{-1} + b^{-2}} = a^{-1} + b^{-1} = \frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab}$

$(\frac{a^3 - b^3}{a^3 + b^3})^5 * (\frac{a^3 - b^3}{a^3 + b^3})^{-5} = (\frac{a^3 - b^3}{a^3 + b^3})^5 * (\frac{a^3 + b^3}{a^3 - b^3})^5 = (\frac{a^3 - b^3}{a^3 + b^3} * \frac{a^3 + b^3}{a^3 - b^3})^5 = 1^5 = 1$

$(\frac{a^2 - a^2}{a^2 + a^{-2}})^7 : (\frac{a^2 + a^{-2}}{a^2 - a^{-2}})^{-7} = (\frac{a^2 - a^{-2}}{a^2 + a^{-2}})^7 : (\frac{a^2 - a^{-2}}{a^2 - a^{-2}})^7 = (\frac{a^2 - a^{-2}}{a^2 + a^{-2}})^7 * (\frac{a^2 + a^{-2}}{a^2 - a^{-2}})^7 = (\frac{a^2 - a^{-2}}{a^2 + a^{-2}} * \frac{a^2 + a^{-2}}{a^2 - a^{-2}})^7 = 1^7 = 1$

$\frac{\frac{1}{a^2} + \frac{2}{ab} + \frac{1}{b^2}}{\frac{1}{a^2} - \frac{1}{b^2}} = \frac{\frac{b^2 + 2ab + a^2}{a^2b^2}}{\frac{b^2 - a^2}{a^2b^2}} = \frac{(b + a)^2}{a^2b^2} : \frac{(b - a)(b + a)}{a^2b^2} = \frac{(b + a)^2}{a^2b^2} * \frac{a^2b^2}{(b - a)(b + a)} = \frac{b + a}{b - a}$

$\frac{\frac{1}{a^2} - \frac{2}{ab} + \frac{1}{b^2}}{\frac{1}{a^3} - \frac{3}{a^2b} + \frac{3}{ab^2} - \frac{1}{b^3}} = \frac{\frac{b^2 - 2ab + a^2}{a^2b^2}}{\frac{b^3 - 3ab^2 + 3a^2b - a^3}{a^3b^3}} = \frac{(b - a)^2}{a^2b^2} : \frac{(b - a)^3}{a^3b^3} = \frac{(b - a)^2}{a^2b^2} * \frac{a^3b^3}{(b - a)^3} = \frac{ab}{b - a}$