What is the simplest form of \( \frac{x^{4} y^{7}}{\sqrt[3]{x^{10} y^{4}}} ? \)

To simplify \( \frac{x^{4} y^{7}}{\sqrt[3]{x^{10} y^{4}}} \), we start by simplifying the denominator. The cube root can be distributed over the product inside it: \[ \sqrt[3]{x^{10} y^{4}} = \sqrt[3]{x^{10}} \cdot \sqrt[3]{y^{4}} = x^{10/3} \cdot y^{4/3} \] Now, we can rewrite the expression: \[ \frac{x^{4} y^{7}}{x^{10/3} y^{4/3}} = \frac{x^{4}}{x^{10/3}} \cdot \frac{y^{7}}{y^{4/3}} \] Next, we simplify each part. For the \( x \) term, we subtract the exponents: \[ \frac{x^{4}}{x^{10/3}} = x^{4 - 10/3} = x^{12/3 - 10/3} = x^{2/3} \] For the \( y \) term, we also subtract the exponents: \[ \frac{y^{7}}{y^{4/3}} = y^{7 - 4/3} = y^{21/3 - 4/3} = y^{17/3} \] Putting it all together, we have: \[ \frac{x^{4} y^{7}}{\sqrt[3]{x^{10} y^{4}}} = x^{2/3} y^{17/3} \] Thus, the simplest form is: \[ \boxed{x^{\frac{2}{3}} y^{\frac{17}{3}}} \]