巧算. $$\left(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}\right) \times \left(\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}\right) - \left(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}\right) \times \left(\frac{1}{3}+\frac{1}{5}+\frac{1}{7}\right)$$

1. 设未知数换元，将复杂算式转化为字母代数式

设 $a = \frac{1}{3} + \frac{1}{5} + \frac{1}{7}$。
则原式中的各个括号可表示为：
$1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} = 1 + a$
$\frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} = a + \frac{1}{9}$
$1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} = 1 + a + \frac{1}{9}$

2. 代入代数式展开并化简求值

将换元后的表示代入原式：
$\text{原式} = (1 + a)\left(a + \frac{1}{9}\right) - \left(1 + a + \frac{1}{9}\right)a$

利用乘法分配律展开：
$= \left(a + \frac{1}{9} + a^2 + \frac{1}{9}a\right) - \left(a + a^2 + \frac{1}{9}a\right)$
$= a^2 + \frac{10}{9}a + \frac{1}{9} - a^2 - \frac{10}{9}a$
$= \frac{1}{9}$