Use rational exponents to simplify the expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers. \( \frac{12}{x} \) \( \sqrt[13]{x} \)

To simplify the expression \( \frac{12}{x} \cdot \sqrt[13]{x} \), first, rewrite the square root in terms of rational exponents. Recall that \( \sqrt[13]{x} = x^{1/13} \). Thus, the expression becomes \( \frac{12}{x} \cdot x^{1/13} \). Now, rewrite \( \frac{12}{x} \) as \( 12 \cdot x^{-1} \). This gives \( 12 \cdot x^{-1} \cdot x^{1/13} = 12 \cdot x^{-1 + 1/13} \). Simplifying the exponent, we get \( -1 + \frac{1}{13} = -\frac{13}{13} + \frac{1}{13} = -\frac{12}{13} \). So now we have \( 12 \cdot x^{-\frac{12}{13}} \). Finally, convert the negative exponent back to a radical form: \( 12 \cdot \frac{1}{x^{\frac{12}{13}}} = 12 \cdot \frac{1}{\sqrt[13]{x^{12}}} = \frac{12}{\sqrt[13]{x^{12}}} \). Thus, the final simplified expression is \( \frac{12}{\sqrt[13]{x^{12}}} \).