Помогите упростить выражение: [latex] \frac{(ab^{-1}+ a^{-1}b +1)( a^{-1}- b^{-1})^2 }{a^2 b^{-2}+ a^{-2}b^2-(a b^{-1}+ a^{-1}b)} [/latex]

[latex] \frac{(ab^{-1}+a^{-1}b +1)( a^{-1}-b^{-1})^2 }{a^2b^{-2}+a^{-2}b^2-(a b^{-1}+ a^{-1}b)} = \frac{(\frac{a}{b}+\frac{b}{a}+1)(\frac{1}{a}-\frac{1}{b})^2}{\frac{a^2}{b^2}+\frac{b^2}{a^2}-\frac{a}{b}-\frac{b}{a}} =\frac{\frac{a^2+b^2+ab}{ab}\cdot\frac{(b-a)^2}{a^2b^2}}{\frac{a^4+b^4-a^3b-ab^3}{a^2b^2}} = \\ = \frac{(a^2+ab+b^2)(b-a)^2a^2b^2}{a^3b^3(a^3(a-b)-b^3(a-b))} = \frac{(a^2+ab+b^2)(a-b)^2}{ab(a-b)(a^3-b^3)} = \frac{(a^2+ab+b^2)(a-b)^2}{ab(a-b)(a-b)(a^2+ab+b^2)} = [/latex] [latex] \frac{(a^2+ab+b^2)(a-b)^2}{ab(a-b)^2(a^2+ab+b^2)} = \\ = \frac{1}{ab} = a^{-1}b^{-1}[/latex]