典例2 通分：
(1)$\frac{1}{a^2b}, -\frac{2}{ab^2}$；
(2)$\frac{1}{x^2 - y^2}, \frac{1}{x^2 + xy}$。

(1) 最简公分母是$a^2b^2.$
$\frac{1}{a^2b} = \frac{b}{a^2b^2}, -\frac{2}{ab^2} = -\frac{2a}{a^2b^2}.$
(2) 最简公分母是$x(x+y)(x-y).$
$\frac{1}{x^2 - y^2} = \frac{x}{x(x+y)(x-y)} = \frac{x}{x^3 - xy^2},$
$\frac{1}{x^2 + xy} = \frac{x-y}{x(x+y)(x-y)} = \frac{x-y}{x^3 - xy^2}.$

举一反三 通分：
(1) $\frac{1}{4a^2}, \frac{b}{2ac};$
(2) $\frac{2}{9-3a}, \frac{a-1}{a^2 -9}.$
$\frac{1}{a^2b} = \frac{b}{a^2b^2}, -\frac{2}{ab^2} = -\frac{2a}{a^2b^2}.$
(2)最简公分母是$x(x+y)(x-y).$
$\frac{1}{x^2 - y^2} = \frac{x}{x(x+y)(x-y)} = \frac{x}{x^3 - xy^2},$
$\frac{1}{x^2 + xy} = \frac{x-y}{x(x+y)(x-y)} = \frac{x-y}{x^3 - xy^2}.$

(1) 最简公分母是$4a^2c.$
$\frac{1}{4a^2} = \frac{1 · c}{4a^2 · c} = \frac{c}{4a^2c},$
$\frac{b}{2ac} = \frac{b · 2a}{2ac · 2a} = \frac{2ab}{4a^2c}.$
(2) 最简公分母是$3(a+3)(a-3).$
$\frac{2}{9-3a} = -\frac{2}{3(a-3)}$
$= -\frac{2 · (a+3)}{3(a-3) · (a+3)}$
$= -\frac{2a+6}{3a^2 -27},$
$\frac{a-1}{a^2 -9} = \frac{a-1}{(a+3)(a-3)}$
$= \frac{(a-1) · 3}{(a+3)(a-3) · 3}$
$= \frac{3a-3}{3a^2 -27}.$