12.7 Упростить выражение: 1) (x-1-x-3)²+; 2 x4 2) (m-2+m-3)²-2; m³ 3) (x-1-y-¹): (x-2-y-2); 4) (m-2-n-2): (m-1 - n-1); 5) (a-2 + a¯¹ + 1) (a² - a) + a¯¹; 6) (4mn-m¹n): (2n-1-m-¹).

1) $$(x^{-1}-x^{-3})^2 + \frac{2}{x^4} = (\frac{1}{x} - \frac{1}{x^3})^2 + \frac{2}{x^4} = (\frac{x^2 - 1}{x^3})^2 + \frac{2}{x^4} = \frac{x^4 - 2x^2 + 1}{x^6} + \frac{2}{x^4} = \frac{x^4 - 2x^2 + 1 + 2x^2}{x^6} = \frac{x^4 + 1}{x^6}$$.

2) $$(m^{-2}+m^{-3})^2 - \frac{2}{m^3} = (\frac{1}{m^2} + \frac{1}{m^3})^2 - \frac{2}{m^3} = (\frac{m + 1}{m^3})^2 - \frac{2}{m^3} = \frac{m^2 + 2m + 1}{m^6} - \frac{2}{m^3} = \frac{m^2 + 2m + 1 - 2m^3}{m^6}$$.

3) $$(x^{-1} - y^{-1}) : (x^{-2} - y^{-2}) = (\frac{1}{x} - \frac{1}{y}) : (\frac{1}{x^2} - \frac{1}{y^2}) = \frac{y - x}{xy} : \frac{y^2 - x^2}{x^2y^2} = \frac{y - x}{xy} \cdot \frac{x^2y^2}{y^2 - x^2} = \frac{y - x}{xy} \cdot \frac{x^2y^2}{(y - x)(y + x)} = \frac{xy}{y + x}$$.

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

5) $$(a^{-2} + a^{-1} + 1) \cdot (a^2 - a) + a^{-1} = (\frac{1}{a^2} + \frac{1}{a} + 1) \cdot (a^2 - a) + \frac{1}{a} = \frac{1 + a + a^2}{a^2} \cdot a(a - 1) + \frac{1}{a} = \frac{(1 + a + a^2)(a - 1)}{a} + \frac{1}{a} = \frac{a - 1 + a^2 - a + a^3 - a^2 + 1}{a} = \frac{a^3}{a} = a^2$$.

6) $$(4mn^{-1} - m^{-1}n) : (2n^{-1} - m^{-1}) = (\frac{4m}{n} - \frac{n}{m}) : (\frac{2}{n} - \frac{1}{m}) = (\frac{4m^2 - n^2}{mn}) : (\frac{2m - n}{mn}) = \frac{4m^2 - n^2}{mn} \cdot \frac{mn}{2m - n} = \frac{(2m - n)(2m + n)}{mn} \cdot \frac{mn}{2m - n} = 2m + n$$.

Ответ: смотри решение