Use rational exponents to simplify and write the following using a single radical. If \[ \sqrt[3]{u^{3}} \cdot \sqrt[7]{u^{2}}=\sqrt[5]{u^{m_{3}}} \] then \( n=\square \) and \( m=\square \)

To simplify the expression \(\sqrt[3]{u^{3}} \cdot \sqrt[7]{u^{2}}\), we can convert the roots into rational exponents: \[ \sqrt[3]{u^{3}} = u^{3/3} = u^{1}, \quad \text{and} \quad \sqrt[7]{u^{2}} = u^{2/7} \] Next, we can multiply these two expressions together: \[ u^{1} \cdot u^{2/7} = u^{1 + 2/7} = u^{7/7 + 2/7} = u^{9/7} \] Now, we can express this in terms of a single radical: \[ u^{9/7} = \sqrt[7]{u^{9}} \quad \text{(since it is the same as } u^{9/7} = u^{\frac{9}{7}} \text{)} \] Now, we compare this to \(\sqrt[5]{u^{m_{3}}}\). To express it in the same form, we need to equate the exponents: \[ \sqrt[5]{u^{m_{3}}} = u^{m_{3}/5} \quad \Rightarrow \quad m_{3}/5 = 9/7 \] To solve for \(m_{3}\), we multiply both sides by 5: \[ m_{3} = 5 \cdot \frac{9}{7} = \frac{45}{7} \] Therefore, \(m_{3} = \frac{45}{7}\). Now, to find \(n\), we look at the first radical: \[ \sqrt[3]{u^{3}} \implies n = 3 \] So we have: \( n = 3 \) and \( m = \frac{45}{7} \).