Упростите выражение:
1) $\frac{4b}{a^2 - b^2} + \frac{a - b}{a^2 + ab} + \frac{a + b}{b^2 - ab}$;
2) $\frac{1}{x - 2} + \frac{1}{x + 2} - \frac{x}{x^2 - 4} + \frac{x^2 + 4}{8x - 2x^3}$;
3) $\frac{1}{(a - 5b)^2} - \frac{2}{a^2 - 25b^2} + \frac{1}{(a + 5b)^2}$;
4) $\frac{x^2 + 9x + 18}{xy + 3y - 2x - 6} - \frac{x + 5}{y - 2}$.
$\frac{4b}{a^2 - b^2} + \frac{a - b}{a^2 + ab} + \frac{a + b}{b^2 - ab} = \frac{4b}{a^2 - b^2} + \frac{a - b}{a^2 + ab} - \frac{a + b}{ab - b^2} = \frac{4b}{(a - b)(a + b)} + \frac{a - b}{a(a + b)} - \frac{a + b}{b(a - b)} = \frac{4b * ab + b(a - b)(a - b) - a(a + b)(a + b)}{ab(a - b)(a + b)} = \frac{4ab^2 + b(a - b)^2 - a(a + b)^2}{ab(a - b)(a + b)} = \frac{4ab^2 + b(a^2 - 2ab + b^2) - a(a^2 + 2ab + b^2)}{ab(a - b)(a + b)} = \frac{4ab^2 + a^2b - 2ab^2 + b^3 - a^3 - 2a^2b - ab^2}{ab(a - b)(a + b)} = \frac{ab^2 - a^2b + b^3 - a^3}{ab(a - b)(a + b)} = \frac{(ab^2 - a^2b) + (b^3 - a^3)}{ab(a - b)(a + b)} = \frac{ab(b - a) + (b - a)(b^2 + ab + a^2)}{ab(a - b)(a + b)} = \frac{(b - a)(ab + b^2 + ab + a^2}{ab(a - b)(a + b)} = \frac{-(a - b)(a^2 + 2ab + b^2)}{ab(a - b)(a + b)} = \frac{-(a - b)(a + b)^2}{ab(a - b)(a + b)} = \frac{-(a + b)}{ab} = -\frac{a + b}{ab}$
$\frac{1}{x - 2} + \frac{1}{x + 2} - \frac{x}{x^2 - 4} + \frac{x^2 + 4}{8x - 2x^3} = \frac{1}{x - 2} + \frac{1}{x + 2} - \frac{x}{x^2 - 4} - \frac{x^2 + 4}{2x^3 - 8x} = \frac{1}{x - 2} + \frac{1}{x + 2} - \frac{x}{(x - 2)(x + 2)} - \frac{x^2 + 4}{2x(x^2 - 4)} = \frac{1}{x - 2} + \frac{1}{x + 2} - \frac{x}{(x - 2)(x + 2)} - \frac{x^2 + 4}{2x(x - 2)(x + 2)} = \frac{2x(x + 2) + 2x(x - 2) - 2x * x - (x^2 + 4)}{2x(x - 2)(x + 2)} = \frac{2x^2 + 4x + 2x^2 - 4x - 2x^2 - x^2 - 4}{2x(x - 2)(x + 2)} = \frac{x^2 - 4}{2x(x^2 - 4)} = \frac{1}{2x}$
$\frac{1}{(a - 5b)^2} - \frac{2}{a^2 - 25b^2} + \frac{1}{(a + 5b)^2} = \frac{1}{(a - 5b)^2} - \frac{2}{(a - 5b)(a + 5b)} + \frac{1}{(a + 5b)^2} = \frac{(a + 5b)^2 - 2(a^2 - 25b^2) + (a - 5b)^2}{(a - 5b)^2(a + 5b)^2} = \frac{a^2 + 10ab + 25b^2 - 2a^2 + 50b^2 + a^2 - 10ab + 25b^2}{(a - 5b)^2(a + 5b)^2} = \frac{100b^2}{(a^2 - 25b^2)^2}$
$\frac{x^2 + 9x + 18}{xy + 3y - 2x - 6} - \frac{x + 5}{y - 2} = \frac{x^2 + 9x + 18}{(xy + 3y) - (2x + 6)} - \frac{x + 5}{y - 2} = \frac{x^2 + 9x + 18}{y(x + 3) - 2(x + 3)} - \frac{x + 5}{y - 2} = \frac{x^2 + 9x + 18}{(x + 3)(y - 2)} - \frac{x + 5}{y - 2} = \frac{x^2 + 9x + 18 - (x + 3)(x + 5)}{(x + 3)(y - 2)} = \frac{x^2 + 9x + 18 - (x^2 + 3x + 5x + 15)}{(x + 3)(y - 2)} = \frac{x^2 + 9x + 18 - x^2 - 3x - 5x - 15}{(x + 3)(y - 2)} = \frac{x + 3}{(x + 3)(y - 2)} = \frac{1}{y - 2}$
Пожауйста, оцените решение
