3. \( \left(\frac{m^{4} \cdot n^{39}}{m^{3} \cdot n^{5}}\right) \cdot\left(\frac{m^{5}}{n^{4}}\right)^{2} \)

To simplify the expression \( \left(\frac{m^{4} \cdot n^{39}}{m^{3} \cdot n^{5}}\right) \cdot\left(\frac{m^{5}}{n^{4}}\right)^{2} \), we can break it down step by step. 1. First, simplify the fraction on the left: \[ \frac{m^{4} \cdot n^{39}}{m^{3} \cdot n^{5}} = \frac{m^{4}}{m^{3}} \cdot \frac{n^{39}}{n^{5}} = m^{4-3} \cdot n^{39-5} = m^{1} \cdot n^{34} = m \cdot n^{34}. \] 2. Now, simplify the right part \(\left( \frac{m^{5}}{n^{4}} \right)^{2}\): \[ \left( \frac{m^{5}}{n^{4}} \right)^{2} = \frac{(m^{5})^{2}}{(n^{4})^{2}} = \frac{m^{10}}{n^{8}}. \] 3. Now, we combine both parts: \[ (m \cdot n^{34}) \cdot \left( \frac{m^{10}}{n^{8}} \right) = \frac{m^{1} \cdot n^{34} \cdot m^{10}}{n^{8}} = \frac{m^{1 + 10}}{n^{8}} \cdot n^{34} = \frac{m^{11}}{n^{8}} \cdot n^{34} = m^{11} \cdot n^{34 - 8} = m^{11} \cdot n^{26}. \] The final simplified expression is: \[ m^{11} \cdot n^{26}. \]