With a brief calculation, determine the value of the following product:

$$
\frac{6 \cdot 27^{12}+2 \cdot 81^{9}}{8000000^{2}} \cdot \frac{80 \cdot 32^{3} \cdot 125^{4}}{9^{19}-729^{6}}
$$
Factoring into prime factors and swapping the two denominators:

$$
\begin{gathered}
\frac{2 \cdot 3 \cdot\left(3^{3}\right)^{12}+2 \cdot\left(3^{4}\right)^{9}}{\left(3^{2}\right)^{19}-\left(3^{6}\right)^{6}} \cdot \frac{2^{4} \cdot 5 \cdot\left(2^{5}\right)^{3} \cdot\left(5^{3}\right)^{4}}{\left(2^{3} \cdot 2^{6} \cdot 5^{6}\right)^{2}}=\frac{2 \cdot 3^{37}+2 \cdot 3^{36}}{3^{38}-3^{36}} \cdot \frac{2^{19} \cdot 5^{13}}{2^{38} \cdot 5^{12}}= \\
=\frac{2 \cdot 3^{36} \cdot(3+1)}{3^{36} \cdot\left(3^{2}-1\right)} \cdot 2 \cdot 5=10
\end{gathered}
$$

Gaganetz Sarolta (Miskolc, Teacher Training College I. o. t.)