Separate The Wheat

By Heron's Formula, we have that the area $S$ of the triangle with sides $a,b,c$ can be represented by $16S^2 = (a+b+c)(a+b-c)(a+c-b)(b+c-a)$ .

For the first numbers, see that $\left ( \dfrac{1}{168}; \dfrac{1}{175}; \dfrac{1}{600} \right ) = \left ( \dfrac{25}{7 \cdot 24 \cdot 25}; \dfrac{24}{7 \cdot 24 \cdot 25}; \dfrac{7}{7 \cdot 24 \cdot 25} \right )$ .

Because $7^2 + 24^2 = 25^2$ , we have that $S_{A} = \dfrac{7 \cdot 24}{2 \cdot (7 \cdot 24 \cdot 25)^2} \Leftrightarrow \mathbb{A} = 16{ \left ( \dfrac{7 \cdot 24}{2 \cdot (7 \cdot 24 \cdot 25)^2} \right ) }^2 = \dfrac{1}{210000^2}$ .

Similarly, for the second number we have $240 = 10 \cdot 24, \; 260 = 10 \cdot 26, \; 624 = 24 \cdot 26$ and $\mathbb{B} = \dfrac{1}{324480^2}$ . For the third number, check $432 = 18 \cdot 24, \; 540 = 18 \cdot 30, \; 720 = 24 \cdot 30$ and $\mathbb{C} = \dfrac{1}{777600^2}.$

Finally, $\sqrt{\dfrac{1}{\mathbb{A}}} = 210000, \; \sqrt{\dfrac{1}{\mathbb{B}}} = 324480, \; \sqrt{\dfrac{1}{\mathbb{C}}} = 777600$ , so $210000 + 324480 + 777600 = 1312080$ . and our desired answer is $1+3+1+2+8 = \boxed{15}$ .