To find the simplified product of \(\sqrt{2 x^3} \cdot \sqrt{18 x^5}\), follow these steps:
### Step 1: Multiply the Radicals
First, we need to multiply the two square root expressions together:
[tex]\[
\sqrt{2 x^3} \cdot \sqrt{18 x^5}
\][/tex]
### Step 2: Combine the Radicands
Under the properties of square roots (i.e., \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)), we can combine the expressions under a single square root:
[tex]\[
\sqrt{2 x^3 \cdot 18 x^5}
\][/tex]
### Step 3: Multiply Inside the Square Root
Now, let's multiply the expressions inside the square root:
[tex]\[
2 x^3 \cdot 18 x^5 = (2 \cdot 18) \cdot (x^3 \cdot x^5)
\][/tex]
[tex]\[
= 36 \cdot x^{3+5}
\][/tex]
[tex]\[
= 36 x^8
\][/tex]
### Step 4: Simplify the Square Root
Next, we take the square root of the combined expression:
[tex]\[
\sqrt{36 x^8}
\][/tex]
### Step 5: Break Down the Components
We can simplify this further by taking the square root of each component separately:
[tex]\[
\sqrt{36 x^8} = \sqrt{36} \cdot \sqrt{x^8}
\][/tex]
[tex]\[
= 6 \cdot x^4
\][/tex]
This is because the square root of \(36\) is \(6\), and the square root of \(x^8\) is \(x^4\).
### Conclusion
The simplified product of \(\sqrt{2 x^3} \cdot \sqrt{18 x^5}\) is \(6 x^4\).
Thus, the correct answer is:
[tex]\[
6 x^4
\][/tex]
