Express the answer in simplest radical form. \[ 8 \sqrt[3]{27 x^{4} y^{3}} \]

To simplify the expression \( 8 \sqrt[3]{27 x^{4} y^{3}} \), we start by breaking it down: The cube root can be separated into individual components: \[ \sqrt[3]{27} = 3 \quad \text{(since \( 27 = 3^3 \))} \] Next, for the variables: \[ \sqrt[3]{x^4} = x^{4/3} = x^{1 + 1/3} = x \cdot \sqrt[3]{x} \] \[ \sqrt[3]{y^3} = y \quad \text{(since \( 3 \) is a perfect cube)} \] Now we can combine these results: \[ \sqrt[3]{27 x^{4} y^{3}} = 3 \cdot \left(x \cdot \sqrt[3]{x}\right) \cdot y = 3xy \sqrt[3]{x} \] Substituting this back into the original expression yields: \[ 8 \sqrt[3]{27 x^{4} y^{3}} = 8 \cdot (3xy \sqrt[3]{x}) = 24xy \sqrt[3]{x} \] Thus, the simplest radical form of \( 8 \sqrt[3]{27 x^{4} y^{3}} \) is: \[ \boxed{24xy \sqrt[3]{x}} \]