Question

17．三个有理数a，b，c，满足abc＞0，求$\frac{a}{|\begin{array}{l}{a}\\{\;}\end{array}|}+\frac{b}{|\begin{array}{l}{b}\\{\;}\end{array}|}+\frac{c}{|\begin{array}{l}{c}\\{\;}\end{array}|}$的值．

Answer 1

分析 根据题意，由于abc＞0，依据有理数乘法法则分类讨论即可．

解答 解：由abc＞0可得出：
（1）a为负，则b、c异号，负因数有2个，$\frac{a}{|\begin{array}{l}{a}\\{\;}\end{array}|}+\frac{b}{|\begin{array}{l}{b}\\{\;}\end{array}|}+\frac{c}{|\begin{array}{l}{c}\\{\;}\end{array}|}$=-1-1+1=-1；
（2）b为负，则a、c异号，负因数有2个，$\frac{a}{|\begin{array}{l}{a}\\{\;}\end{array}|}+\frac{b}{|\begin{array}{l}{b}\\{\;}\end{array}|}+\frac{c}{|\begin{array}{l}{c}\\{\;}\end{array}|}$=-1-1+1=-1；
（3）c为负，则a、b异号，负因数有2个，$\frac{a}{|\begin{array}{l}{a}\\{\;}\end{array}|}+\frac{b}{|\begin{array}{l}{b}\\{\;}\end{array}|}+\frac{c}{|\begin{array}{l}{c}\\{\;}\end{array}|}$=-1-1+1=-1；
（4）c为正，则a、b为正，负因数有0个，$\frac{a}{|\begin{array}{l}{a}\\{\;}\end{array}|}+\frac{b}{|\begin{array}{l}{b}\\{\;}\end{array}|}+\frac{c}{|\begin{array}{l}{c}\\{\;}\end{array}|}$=1+1+1=3；
所以$\frac{a}{|\begin{array}{l}{a}\\{\;}\end{array}|}+\frac{b}{|\begin{array}{l}{b}\\{\;}\end{array}|}+\frac{c}{|\begin{array}{l}{c}\\{\;}\end{array}|}$的值是-1或3．

点评 本题考查了有理数的乘法，几个不等于零的数相乘，积的符号由负因数的个数决定：当负因数有奇数个数，积为负；当负因数的个数为偶数个时，积为正．