Simplify \( \left(\frac{a^{2}}{b^{3}}\right)^{5}\left(\frac{b}{a^{5}}\right)^{2} \) \( \frac{[?]}{\left.b^{[ }\right]} \)

To simplify the expression \( \left(\frac{a^{2}}{b^{3}}\right)^{5}\left(\frac{b}{a^{5}}\right)^{2} \), we apply the power rule: \((\frac{x}{y})^n = \frac{x^n}{y^n}\). First, simplify each part: 1. \( \left(\frac{a^{2}}{b^{3}}\right)^{5} = \frac{a^{2 \cdot 5}}{b^{3 \cdot 5}} = \frac{a^{10}}{b^{15}} \) 2. \( \left(\frac{b}{a^{5}}\right)^{2} = \frac{b^{2}}{a^{10}} \) Now, combine the two results: \[ \frac{a^{10}}{b^{15}} \cdot \frac{b^{2}}{a^{10}} = \frac{a^{10} \cdot b^{2}}{b^{15} \cdot a^{10}} \] The \( a^{10} \) terms cancel out: \[ = \frac{b^{2}}{b^{15}} = b^{2 - 15} = b^{-13} \] Thus, in the final form, we write it as: \[ \frac{1}{b^{13}} \] So, the answer is: \(\frac{1}{b^{13}} \)