To simplify the expression \((\frac{y^5}{y^3})^2 \times (\frac{y^7}{y^5})^5\), we can start by simplifying each part of the expression separately.
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Simplifying \((\frac{y^5}{y^3})^2\): \[ \frac{y^5}{y^3} = y^{5-3} = y^2 \] Therefore, \[ \left(\frac{y^5}{y^3}\right)^2 = (y^2)^2 = y^{2 \cdot 2} = y^4. \]
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Simplifying \((\frac{y^7}{y^5})^5\): \[ \frac{y^7}{y^5} = y^{7-5} = y^2 \] Therefore, \[ \left(\frac{y^7}{y^5}\right)^5 = (y^2)^5 = y^{2 \cdot 5} = y^{10}. \]
Now, combine the results from both simplifications: \[ y^4 \times y^{10} = y^{4+10} = y^{14}. \]
The final simplified expression is: \[ \boxed{y^{14}}. \]