`sqrt(x^2 y^3 + 3x^2y^3)`, assuming x and y are positive

Answer

To solve the expression \( \sqrt{x^2 y^3 + 3x^2y^3} \), we can follow these steps:

### Step 1: Factor out the common term
We notice that both terms inside the square root share a common factor of \( x^2y^3 \). So, we can factor this out:

\[ \sqrt{x^2 y^3 + 3x^2 y^3} = \sqrt{x^2 y^3 (1 + 3)} \]

### Step 2: Simplify the expression inside the square root
Now we simplify the expression inside the square root:

\[ \sqrt{x^2 y^3 (1 + 3)} = \sqrt{x^2 y^3 \cdot 4} \]

### Step 3: Break down the square root
Next, we can separate the square root of the product:

\[ \sqrt{x^2 y^3 \cdot 4} = \sqrt{x^2} \cdot \sqrt{y^3} \cdot \sqrt{4} \]

### Step 4: Calculate the square roots
Now we calculate the square roots of each component:

\[ \sqrt{x^2} = x, \quad \sqrt{y^3} = y^{3/2} = y \cdot \sqrt{y}, \quad \text{and } \sqrt{4} = 2 \]

### Step 5: Combine the results
Putting it all together, we have:

\[ \sqrt{x^2 y^3 + 3x^2y^3} = x \cdot (y \cdot \sqrt{y}) \cdot 2 = 2xy\sqrt{y} \]

Thus, the final answer is:

\[ \sqrt{x^2 y^3 + 3x^2y^3} = 2xy\sqrt{y} \]

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