Use ratienal expenents to simplify and write the following using a single radical, If \[ \sqrt[3]{u^{2}} \cdot \sqrt[4]{u^{2}}=\sqrt[2]{u^{m}} \] then \( n=\square \) and \( m=\square \) (1)

To simplify the expression \(\sqrt[3]{u^{2}} \cdot \sqrt[4]{u^{2}}\), you can convert the radicals into exponential form. This gives you: \[ u^{2/3} \cdot u^{2/4} = u^{2/3} \cdot u^{1/2} \] Now, you add the exponents together, which results in: \[ u^{2/3 + 1/2} = u^{4/6 + 3/6} = u^{7/6} \] Next, you need to match this with the expression \(\sqrt[2]{u^{m}} = u^{m/2}\). Setting the exponents equal to each other, we have: \[ \frac{7}{6} = \frac{m}{2} \] Cross-multiplying gives: \[ 7 \cdot 2 = 6m \implies m = \frac{14}{6} = \frac{7}{3} \] Thus, \(m = \frac{7}{3}\). Now, we need to write the left-hand side using a single radical. We have: \[ u^{7/6} = \sqrt[6]{u^{7}} \] This means \(n = 6\) and \(m = \frac{7}{3}\). Finally, we get: \( n=6 \) and \( m=\frac{7}{3} \).