Решение задания 1081

а) $$xy^{-2} - x^{-2}y$$

$$ x y^{-2} - x^{-2} y = \frac{x}{y^2} - \frac{y}{x^2} = \frac{x^3 - y^3}{x^2 y^2} $$

б) $$\left(\frac{x}{y}\right)^{-1} + \left(\frac{x}{y}\right)^{-2}$$

$$ \left(\frac{x}{y}\right)^{-1} + \left(\frac{x}{y}\right)^{-2} = \frac{y}{x} + \frac{y^2}{x^2} = \frac{xy + y^2}{x^2} = \frac{y(x + y)}{x^2} $$

в) $$mn(n - m)^{-2} – n(m – n)^{-1}$$

$$ mn(n - m)^{-2} - n(m - n)^{-1} = \frac{mn}{(n - m)^2} - \frac{n}{m - n} = \frac{mn}{(n - m)^2} + \frac{n}{n - m} = \frac{mn + n(n - m)}{(n - m)^2} = \frac{mn + n^2 - mn}{(n - m)^2} = \frac{n^2}{(n - m)^2} $$

г) $$(x^{-1}+y^{-1})(x^{-1}-y^{-1})$$

$$ (x^{-1} + y^{-1})(x^{-1} - y^{-1}) = x^{-2} - y^{-2} = \frac{1}{x^2} - \frac{1}{y^2} = \frac{y^2 - x^2}{x^2 y^2} $$