Question

Assuming xx and yy are both positive, write the following expression in simplest radical form.
2, x, squared, y, cubed, square root of, 9, x, cubed, y, squared, end square root
2x
2
y
3

9x
3
y
2

​

Answer 1

To simplify the expression \( 2x^2y^3 \sqrt{9x^3y^2} \), we first simplify the square root part.

### Step 1: Simplifying the square root
\[
\sqrt{9x^3y^2}
\]
We can break this down into components:

\[
\sqrt{9} = 3 \\
\sqrt{x^3} = \sqrt{x^2 \cdot x} = x\sqrt{x} \\
\sqrt{y^2} = y
\]

Putting it all together:

\[
\sqrt{9x^3y^2} = 3 \cdot x\sqrt{x} \cdot y = 3xy\sqrt{x}
\]

### Step 2: Multiplying the expressions
Now we have:

\[
2x^2y^3 \sqrt{9x^3y^2} = 2x^2y^3(3xy\sqrt{x})
\]

### Step 3: Distributing
Now multiply:
\[
= 2 \cdot 3 \cdot x^2 \cdot y^3 \cdot x \cdot y \cdot \sqrt{x}
\]

### Step 4: Combine like terms
\[
= 6x^{2+1}y^{3+1}\sqrt{x} = 6x^3y^4\sqrt{x}
\]

### Final Result
Thus, the expression \( 2x^2y^3 \sqrt{9x^3y^2} \) in simplest radical form is:

\[
\boxed{6x^3y^4\sqrt{x}}
\]