Express the following in simplified radical form. \[ \sqrt{x y^{3}} \sqrt[3]{x^{2} y} \]

Simplify the expression by following steps: - step0: Solution: \(\sqrt{xy^{3}}\) - step1: Rewrite the exponent: \(\sqrt{xy^{2+1}}\) - step2: Expand the expression: \(\sqrt{xy^{2}\times y}\) - step3: Reorder the terms: \(\sqrt{y^{2}xy}\) - step4: Use the properties of radicals: \(\sqrt{y^{2}}\times \sqrt{xy}\) - step5: Simplify the root: \(y\sqrt{yx}\) Let's simplify the expression step by step. We have: \[ \sqrt{x y^{3}} \sqrt[3]{x^{2} y} \] ### Step 1: Simplify each radical separately 1. **Square Root:** \[ \sqrt{x y^{3}} = \sqrt{x} \cdot \sqrt{y^{3}} = \sqrt{x} \cdot y^{3/2} = y^{3/2} \sqrt{x} \] 2. **Cube Root:** \[ \sqrt[3]{x^{2} y} = \sqrt[3]{x^{2}} \cdot \sqrt[3]{y} = x^{2/3} \cdot y^{1/3} \] ### Step 2: Combine the results Now we can combine the two simplified forms: \[ y^{3/2} \sqrt{x} \cdot x^{2/3} \cdot y^{1/3} \] ### Step 3: Combine like terms 1. **Combine the \(y\) terms:** \[ y^{3/2} \cdot y^{1/3} = y^{3/2 + 1/3} \] To add the exponents, we need a common denominator: \[ \frac{3}{2} = \frac{9}{6}, \quad \frac{1}{3} = \frac{2}{6} \quad \Rightarrow \quad y^{\frac{9}{6} + \frac{2}{6}} = y^{\frac{11}{6}} \] 2. **Combine the \(x\) terms:** \[ \sqrt{x} \cdot x^{2/3} = x^{1/2} \cdot x^{2/3} = x^{\frac{1}{2} + \frac{2}{3}} \] Again, we need a common denominator: \[ \frac{1}{2} = \frac{3}{6}, \quad \frac{2}{3} = \frac{4}{6} \quad \Rightarrow \quad x^{\frac{3}{6} + \frac{4}{6}} = x^{\frac{7}{6}} \] ### Final Result Combining everything, we have: \[ \sqrt{x y^{3}} \sqrt[3]{x^{2} y} = x^{\frac{7}{6}} y^{\frac{11}{6}} \] Thus, the expression in simplified radical form is: \[ \boxed{x^{\frac{7}{6}} y^{\frac{11}{6}}} \]