Let's determine the value of the following expression with as few calculations as possible:

$$
\frac{49 \cdot 91^{3}+338 \cdot 343^{2}}{66^{3}-176 \cdot 121}: \frac{39^{3} \cdot 7^{5}}{1331000}
$$
In the expression, we factor the numbers into prime factors and use powers for calculations:

$$
\begin{gathered}
\frac{49 \cdot 91^{3}+338 \cdot 343^{2}}{66^{3}-176 \cdot 121}: \frac{39^{3} \cdot 7^{5}}{1331000}=\frac{7^{2} \cdot 7^{3} \cdot 13^{3}+2 \cdot 13^{2} \cdot 7^{6}}{2^{3} \cdot 3^{3} \cdot 11^{3}-2^{4} \cdot 11 \cdot 11^{2}} \cdot \frac{2^{3} \cdot 5^{3} \cdot 11^{3}}{3^{3} \cdot 13^{3} \cdot 7^{5}}= \\
\frac{7^{5} \cdot 13^{2}(13+2 \cdot 7)}{2^{3} \cdot 11^{3}\left(3^{3}-2\right)} \cdot \frac{2^{3} \cdot 5^{3} \cdot 11^{3}}{3^{3} \cdot 13^{3} \cdot 7^{5}}=\frac{3^{3} \cdot 7^{5} \cdot 13^{2}}{2^{3} \cdot 5^{2} \cdot 11^{3}} \cdot \frac{2^{3} \cdot 5^{3} \cdot 11^{3}}{3^{3} \cdot 13^{3} \cdot 7^{5}}=\frac{5}{13}
\end{gathered}
$$

Attila Macskásy (Kaposvár, Táncsics g. II. o. t.)