Упражнение 256

\(1)\ ab^{{-}1}+a^{{-}1}b=a\cdot\frac{1}{b}+\frac{1}{a}\cdot{b}={\frac{a}{b}}^{\backslash{a}}+{\frac{b}{a}}^{\backslash{b}}=\frac{a^2+b^2}{ab};\)

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

\(3)\ m^2n^2(m^{{-}3}-n^{{-}3})=m^2n^2\left({\frac{1}{m^3}}^{\backslash{n^3}}-{\frac{1}{n^3}}^{\backslash{m^3}}\right)=m^2n^2\cdot\frac{n^3-m^3}{m^3n^3}=\frac{n^3-m^3}{mn};\)

\(4)\ (a+b)^{{-}1}\cdot(a^{{-}1}+b^{{-}1})=\frac{1}{a+b}\cdot\left({\frac{1}{a}}^{\backslash{b}}+{\frac{1}{b}}^{\backslash{a}}\right)=\frac{1}{a+b}\cdot\frac{b+a}{ab}=\frac{1}{ab};\)

\(5)\ (c^{{-}2}-d^{{-}2}):(c+d)=\left({\frac{1}{c^2}}^{\backslash{d^2}}-{\frac{1}{d^2}}^{\backslash{c^2}}\right)\cdot\frac{1}{c+d}=\frac{d^2-c^2}{c^2d^2}\cdot\frac{1}{c+d}=\frac{ (d-c)(d+c)}{c^2d^2}\cdot\frac{1}{c+d}=\frac{d-c}{c^2d^2};\)

\(6)\ (xy^{{-}2}+x^{{-}2}y)\cdot\left(\frac{x^2-xy+y^2}{x}\right)^{{-}1}=\left(x\cdot\frac{1}{y^2}+\frac{1}{x^2}\cdot{y}\right)\cdot\frac{x}{x^2-xy+y^2}=\left({\frac{x}{y^2}}^{\backslash{x^2}}+{\frac{y}{x^2}}^{\backslash{y^2}}\right)\cdot\frac{x}{x^2-xy+y^2}=\frac{x^3+y^3}{x^2y^2}\cdot\frac{x}{x^2-xy+y^2}=\frac{ (x+y)(x^2-xy+y^2)}{x^2y^2}\cdot\frac{x}{x^2-xy+y^2}=\frac{x+y}{xy^2}.\)