Express in simplest radical form. \[ 729 x^{\frac{5}{8}} \]

To express \( 729 x^{\frac{5}{8}} \) in simplest radical form, we start by simplifying each part separately. First, \( 729 \) is a perfect cube, specifically \( 729 = 9^3 = 3^6 \). Hence, we can rewrite \( 729 \) as \( 27^2 \) or \( 3^6 = (3^2)^3 \). Next, we simplify \( x^{\frac{5}{8}} \). The exponent \( \frac{5}{8} \) can be expressed as \( \frac{5}{8} = \frac{1}{8} \cdot 5 \), indicating we have an eighth root involved. Thus, we have: \[ x^{\frac{5}{8}} = \sqrt[8]{x^5} \] Combining these, we get: \[ 729 x^{\frac{5}{8}} = 27^2 \cdot \sqrt[8]{x^5} \] Expressing our findings in radical form, we can present it as: \[ 27 \sqrt[8]{x^5} \] Therefore, the final answer in simplest radical form is: \[ 27 \sqrt[8]{x^5} \]