Конечно, сейчас решим эти выражения!
1) $\displaystyle \left(x + \frac{x}{y}\right) : \left(x - \frac{x}{y}\right)$
* $\displaystyle x + \frac{x}{y} = \frac{xy + x}{y} = \frac{x(y + 1)}{y}$
* $\displaystyle x - \frac{x}{y} = \frac{xy - x}{y} = \frac{x(y - 1)}{y}$
* $\displaystyle \frac{x(y + 1)}{y} : \frac{x(y - 1)}{y} = \frac{x(y + 1)}{y} \cdot \frac{y}{x(y - 1)} = \frac{y + 1}{y - 1}$
**Ответ: $\displaystyle \frac{y + 1}{y - 1}$**
2) $\displaystyle \left(\frac{a}{b} + \frac{a + b}{a - b}\right) \cdot \frac{ab^2}{a^2 + b^2}$
* $\displaystyle \frac{a}{b} + \frac{a + b}{a - b} = \frac{a(a - b) + b(a + b)}{b(a - b)} = \frac{a^2 - ab + ab + b^2}{b(a - b)} = \frac{a^2 + b^2}{b(a - b)}$
* $\displaystyle \frac{a^2 + b^2}{b(a - b)} \cdot \frac{ab^2}{a^2 + b^2} = \frac{ab^2}{b(a - b)} = \frac{ab}{a - b}$
**Ответ: $\displaystyle \frac{ab}{a - b}$**
3) $\displaystyle \left(\frac{m}{m - 1} - 1\right) : \frac{m}{mn - n}$
* $\displaystyle \frac{m}{m - 1} - 1 = \frac{m - (m - 1)}{m - 1} = \frac{m - m + 1}{m - 1} = \frac{1}{m - 1}$
* $\displaystyle \frac{m}{mn - n} = \frac{m}{n(m - 1)}$
* $\displaystyle \frac{1}{m - 1} : \frac{m}{n(m - 1)} = \frac{1}{m - 1} \cdot \frac{n(m - 1)}{m} = \frac{n}{m}$
**Ответ: $\displaystyle \frac{n}{m}$**
4) $\displaystyle \left(\frac{a}{b} - \frac{b}{a}\right) \cdot \frac{4ab}{a - b}$
* $\displaystyle \frac{a}{b} - \frac{b}{a} = \frac{a^2 - b^2}{ab}$
* $\displaystyle \frac{a^2 - b^2}{ab} \cdot \frac{4ab}{a - b} = \frac{(a - b)(a + b)}{ab} \cdot \frac{4ab}{a - b} = 4(a + b)$
**Ответ: $\displaystyle 4(a + b)$**
5) $\displaystyle \frac{a}{b} - \frac{a^2 - b^2}{b^2} : \frac{a + b}{b}$
* $\displaystyle \frac{a^2 - b^2}{b^2} : \frac{a + b}{b} = \frac{(a - b)(a + b)}{b^2} \cdot \frac{b}{a + b} = \frac{a - b}{b}$
* $\displaystyle \frac{a}{b} - \frac{a - b}{b} = \frac{a - (a - b)}{b} = \frac{a - a + b}{b} = \frac{b}{b} = 1$
**Ответ: 1**
6) $\displaystyle \frac{7x}{x + 2} - \frac{x - 8}{3x + 6} \cdot \frac{84}{x^2 - 8x}$
* $\displaystyle \frac{x - 8}{3x + 6} = \frac{x - 8}{3(x + 2)}$
* $\displaystyle \frac{84}{x^2 - 8x} = \frac{84}{x(x - 8)}$
* $\displaystyle \frac{x - 8}{3(x + 2)} \cdot \frac{84}{x(x - 8)} = \frac{84}{3x(x + 2)} = \frac{28}{x(x + 2)}$
* $\displaystyle \frac{7x}{x + 2} - \frac{28}{x(x + 2)} = \frac{7x^2 - 28}{x(x + 2)} = \frac{7(x^2 - 4)}{x(x + 2)} = \frac{7(x - 2)(x + 2)}{x(x + 2)} = \frac{7(x - 2)}{x}$
**Ответ: $\displaystyle \frac{7(x - 2)}{x}$**
7) $\displaystyle \left(a - \frac{9a - 9}{a + 3}\right) : \frac{a^2 - 3a}{a + 3}$
* $\displaystyle a - \frac{9a - 9}{a + 3} = \frac{a(a + 3) - (9a - 9)}{a + 3} = \frac{a^2 + 3a - 9a + 9}{a + 3} = \frac{a^2 - 6a + 9}{a + 3} = \frac{(a - 3)^2}{a + 3}$
* $\displaystyle \frac{a^2 - 3a}{a + 3} = \frac{a(a - 3)}{a + 3}$
* $\displaystyle \frac{(a - 3)^2}{a + 3} : \frac{a(a - 3)}{a + 3} = \frac{(a - 3)^2}{a + 3} \cdot \frac{a + 3}{a(a - 3)} = \frac{a - 3}{a}$
**Ответ: $\displaystyle \frac{a - 3}{a}$**
8) $\displaystyle \left(\frac{a}{a + 2} - \frac{8}{a + 8}\right) \cdot \frac{a^2 + 8a}{a - 4}$
* $\displaystyle \frac{a}{a + 2} - \frac{8}{a + 8} = \frac{a(a + 8) - 8(a + 2)}{(a + 2)(a + 8)} = \frac{a^2 + 8a - 8a - 16}{(a + 2)(a + 8)} = \frac{a^2 - 16}{(a + 2)(a + 8)} = \frac{(a - 4)(a + 4)}{(a + 2)(a + 8)}$
* $\displaystyle \frac{a^2 + 8a}{a - 4} = \frac{a(a + 8)}{a - 4}$
* $\displaystyle \frac{(a - 4)(a + 4)}{(a + 2)(a + 8)} \cdot \frac{a(a + 8)}{a - 4} = \frac{a(a + 4)}{a + 2}$
**Ответ: $\displaystyle \frac{a(a + 4)}{a + 2}$**
