$5.\dot{6}\dot{3} \times 1\frac{13}{31} \times 1.4\dot{9}$ = ___

1. 将循环小数与带分数转化为假分数

将算式中的循环小数和带分数分别化为最简假分数：
$5.\dot{6}\dot{3} = 5 + \frac{63}{99} = 5 + \frac{7}{11} = \frac{62}{11}$
$1\frac{13}{31} = \frac{31 + 13}{31} = \frac{44}{31}$
$1.4\dot{9} = \frac{149 - 14}{90} = \frac{135}{90} = \frac{3}{2}$

2. 代入算式并约分计算乘积

将化简后的假分数代入原式进行乘法计算并约分：
$5.\dot{6}\dot{3} \times 1\frac{13}{31} \times 1.4\dot{9} = \frac{62}{11} \times \frac{44}{31} \times \frac{3}{2}$
$= \left(\frac{62}{31}\right) \times \left(\frac{44}{11}\right) \times \frac{3}{2}$
$= 2 \times 4 \times \frac{3}{2} = 12$