What is the simplest form of \( \frac{\sqrt[3]{81 x^{10}}}{\sqrt[3]{3 x}} \) ?

To simplify the expression \( \frac{\sqrt[3]{81 x^{10}}}{\sqrt[3]{3 x}} \), we can first use the property of cube roots to combine the two terms: \[ \frac{\sqrt[3]{81 x^{10}}}{\sqrt[3]{3 x}} = \sqrt[3]{\frac{81 x^{10}}{3 x}}. \] Next, let's simplify the fraction inside the cube root: \[ \frac{81 x^{10}}{3 x} = \frac{81}{3} \cdot \frac{x^{10}}{x}. \] Calculating \( \frac{81}{3} \): \[ \frac{81}{3} = 27. \] Now simplifying \( \frac{x^{10}}{x} \): \[ \frac{x^{10}}{x} = x^{10-1} = x^9. \] Now we can rewrite our expression: \[ \frac{81 x^{10}}{3 x} = 27 x^9. \] Now substituting this back into the cube root gives us: \[ \sqrt[3]{27 x^9}. \] Next, we simplify \( \sqrt[3]{27} \) and \( \sqrt[3]{x^9} \): \[ \sqrt[3]{27} = 3, \] and for \( \sqrt[3]{x^9} \): \[ \sqrt[3]{x^9} = x^{9/3} = x^3. \] Now combining these results, we have: \[ \sqrt[3]{27 x^9} = 3 x^3. \] Thus, the simplest form of \( \frac{\sqrt[3]{81 x^{10}}}{\sqrt[3]{3 x}} \) is: \[ \boxed{3 x^3}. \]