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

Question

UpStudy AI · Accepted Answer

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

$$
\boxed{y^{2} \sqrt[3]{x}}
$$
The simplified form of the expression is $$ y^{2} \sqrt[3]{x} $$.

B.