B. \( \left(\frac{x y^{3}}{x^{\frac{1}{2}}}\right)^{\frac{2}{3}} \)

Let's simplify the expression: \[ \left(\frac{x y^{3}}{x^{\frac{1}{2}}}\right)^{\frac{2}{3}} \] ### Step 1: Simplify Inside the Parentheses First, simplify the fraction inside the parentheses: \[ \frac{x y^{3}}{x^{\frac{1}{2}}} = x^{1 - \frac{1}{2}} y^{3} = x^{\frac{1}{2}} y^{3} \] ### Step 2: Apply the Exponent Outside the Parentheses Now, raise the simplified expression to the \(\frac{2}{3}\) power: \[ \left(x^{\frac{1}{2}} y^{3}\right)^{\frac{2}{3}} = \left(x^{\frac{1}{2}}\right)^{\frac{2}{3}} \left(y^{3}\right)^{\frac{2}{3}} \] ### Step 3: Simplify Each Term Separately - For \(x^{\frac{1}{2}}\): \[ \left(x^{\frac{1}{2}}\right)^{\frac{2}{3}} = x^{\frac{1}{2} \times \frac{2}{3}} = x^{\frac{1}{3}} \] - For \(y^{3}\): \[ \left(y^{3}\right)^{\frac{2}{3}} = y^{3 \times \frac{2}{3}} = y^{2} \] ### Step 4: Combine the Simplified Terms Putting it all together: \[ x^{\frac{1}{3}} \cdot y^{2} = y^{2} x^{\frac{1}{3}} \] Alternatively, you can write the expression using a radical for the cube root: \[ y^{2} \sqrt[3]{x} \] ### Final Simplified Form \[ \boxed{y^{2} \sqrt[3]{x}} \]