Solve the following system of equations:

$$
\sqrt[3]{7^{x}} \cdot \sqrt[5]{5^{y}}=6125
$$

$$
\sqrt[5]{3^{y}}: \sqrt[3]{3^{x}}=3
$$
The given equations can also be written as follows:

From (2)

which gives

Substituting (3) into (1):

$$
7^{\frac{x}{3}} \cdot 5^{\frac{y}{5}}=6125
$$

or

$$
3^{\frac{y}{5}}: 3^{\frac{x}{3}}=3
$$

$$
\frac{y}{5}-\frac{x}{3}=1
$$

$$
\frac{y}{5}=1+\frac{x}{3}
$$

$$
7^{\frac{x}{3}} \cdot 5^{1+\frac{x}{3}}=6125
$$

$$
7^{\frac{x}{3}} \cdot 5^{\frac{x}{3}}=1225
$$

which gives

$$
35^{\frac{x}{3}}=35^{2}
$$

From (3) we get

$$
\begin{gathered}
x=6 \\
y=15
\end{gathered}
$$

(Manheim Emil, Beszterczebánya.)

The problem was also solved by: Barna D., Beck F., Brandt D., Dénes A., Erdős A., Freibauer E., Friedmann B., Kisgyörgy S., Krátky Gy., Misángyi V., Petrogalli G., Porkoláb J., Schiffer H., Spitzer Ö., Szabó I., Szabó K., Weisz Á., Weisz J.