《2025年阳光课堂金牌练习册八年级数学上册人教版》

【典型例题 1】约分：
(1)$\dfrac{-15x^{2}y}{10xy^{2}}$；(2)$\dfrac{a^{2}+2a+1}{a^{2}-1}$；
(3)$\dfrac{2n^{2}-m}{2mn-4n^{3}}$；(4)$\dfrac{xy^{2}+2y}{y}$.

答案： 【解】
(1)$\dfrac{-15x^{2}y}{10xy^{2}}= -\dfrac{5xy\cdot3x}{5xy\cdot2y}= -\dfrac{3x}{2y}$.
(2)$\dfrac{a^{2}+2a+1}{a^{2}-1}= \dfrac{(a+1)^{2}}{(a+1)(a-1)}= \dfrac{a+1}{a-1}$.
(3)$\dfrac{2n^{2}-m}{2mn-4n^{3}}= \dfrac{-(m-2n^{2})}{2n(m-2n^{2})}= -\dfrac{1}{2n}$.
(4)$\dfrac{xy^{2}+2y}{y}= \dfrac{y(xy+2)}{y}= xy+2$.

1. 化简下列分式：
(1)$\dfrac{-35a^{4}b}{21a^{2}b^{3}}$；(2)$\dfrac{x^{2}+6x+9}{x^{2}-9}$.

答案：
(1)
$\begin{aligned} \dfrac{-35a^{4}b}{21a^{2}b^{3}} &= \dfrac{-35 ÷ 7 \cdot a^{4 - 2} \cdot b^{1 - 3}}{21÷7} \\ &= \dfrac{-5a^{2}}{3b^{2}} \\ &= -\dfrac{5a^{2}}{3b^{2}} \end{aligned}$
(2)
$\begin{aligned} \dfrac{x^{2}+6x + 9}{x^{2}-9} &= \dfrac{(x + 3)^{2}}{(x + 3)(x - 3)} \\ &= \dfrac{x + 3}{x - 3} \end{aligned}$

【典型例题 2】通分：
(1)$\dfrac{3}{4a^{2}b}与\dfrac{1}{2ac^{2}}$；(2)$\dfrac{x+2}{2x+2}与\dfrac{3}{8-4x}$.

答案： 思路导引 通分的关键是确定最简公分母.在确定几个分式的最简公分母时,要从系数、相同因式、不同因式三个方面考虑,不要遗漏只在一个分式的分母中出现的字母及其指数.
【解】
(1)分母$4a^{2}b$,$2ac^{2}的最简公分母是4a^{2}bc^{2}$,$\dfrac{3}{4a^{2}b}= \dfrac{3× c^{2}}{4a^{2}b\cdot c^{2}}= \dfrac{3c^{2}}{4a^{2}bc^{2}}$；
$\dfrac{1}{2ac^{2}}= \dfrac{1×2ab}{2ac^{2}\cdot2ab}= \dfrac{2ab}{4a^{2}bc^{2}}$.
(2)$\dfrac{3}{8-4x}= -\dfrac{3}{4x-8}= -\dfrac{3}{4(x-2)}$,$\dfrac{x+2}{2x+2}= \dfrac{x+2}{2(x+1)}$,分母$4(x-2)$,$2(x+1)的最简公分母是4(x+1)(x-2)$,$\dfrac{x+2}{2x+2}= \dfrac{(x+2)\cdot2(x-2)}{2(x+1)\cdot2(x-2)}= \dfrac{2(x+2)(x-2)}{4(x+1)(x-2)}$；$\dfrac{3}{8-4x}= -\dfrac{3}{4x-8}= -\dfrac{3×(x+1)}{4(x-2)\cdot(x+1)}= -\dfrac{3(x+1)}{4(x+1)(x-2)}$.