Multiply. Assume that the variables represent positive real numbers. \[ \sqrt[8]{9 p^{3} l^{4}} \cdot \sqrt[6]{6 p l^{5}}= \] \( \square \)

To multiply these roots, we can rewrite them using fractional exponents. The expression becomes: \[ (9 p^{3} l^{4})^{1/8} \cdot (6 p l^{5})^{1/6} = 9^{1/8} p^{3/8} l^{4/8} \cdot 6^{1/6} p^{1/6} l^{5/6}. \] Now we combine the like terms: \[ = 9^{1/8} \cdot 6^{1/6} \cdot p^{3/8 + 1/6} \cdot l^{1/2 + 5/6} \] Next, we need to find common denominators for the exponents of \( p \) and \( l \): - The common denominator for \( \frac{3}{8} \) and \( \frac{1}{6} \) is 24: \(\frac{3}{8} = \frac{9}{24}\) and \(\frac{1}{6} = \frac{4}{24}\), so it becomes \( p^{\frac{9 + 4}{24}} = p^{\frac{13}{24}} \). - The common denominator for \( \frac{1}{2} \) and \( \frac{5}{6} \) is 6: \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{5}{6}\) is already in that term, so it becomes \( l^{\frac{3 + 5}{6}} = l^{\frac{8}{6}} = l^{\frac{4}{3}} \). Putting it all together: \[ = 9^{1/8} \cdot 6^{1/6} \cdot p^{\frac{13}{24}} \cdot l^{\frac{4}{3}}. \] Thus, the final answer, in simplified form, is: \[ 9^{1/8} \cdot 6^{1/6} \cdot p^{\frac{13}{24}} \cdot l^{\frac{4}{3}}. \]