To solve \((y^{\frac{3}{5}})^{\frac{10}{15}}\), we can use the property of exponents that states \((a^m)^n = a^{m \cdot n}\).
Here, we have:
\[ (y^{\frac{3}{5}})^{\frac{10}{15}} = y^{\frac{3}{5} \cdot \frac{10}{15}} \]
Next, we simplify \(\frac{10}{15}\):
\[ \frac{10}{15} = \frac{2}{3} \]
Now we multiply the exponents:
\[ \frac{3}{5} \cdot \frac{2}{3} = \frac{3 \cdot 2}{5 \cdot 3} = \frac{6}{15} \]
So, our expression simplifies to:
\[ y^{\frac{6}{15}} \]
Lastly, we can simplify \(\frac{6}{15}\):
\[ \frac{6}{15} = \frac{2}{5} \]
Thus, the final result is:
\[ y^{\frac{2}{5}} \]