To find the simplified product of the expressions [tex]\(\sqrt{2 x^3}\)[/tex] and [tex]\(\sqrt{18 x^3}\)[/tex], we can follow these steps:
1. Rewrite the expressions under a single radical:
[tex]\[
\sqrt{2 x^3} \cdot \sqrt{18 x^3} = \sqrt{(2 x^3) \cdot (18 x^3)}
\][/tex]
2. Combine the terms inside the radical:
[tex]\[
(2 x^3) \cdot (18 x^3) = 2 \cdot 18 \cdot x^3 \cdot x^3 = 36 \cdot x^6
\][/tex]
3. Simplify the combined expression:
[tex]\[
\sqrt{36 x^6}
\][/tex]
4. Separate the constant and variable parts inside the radical:
[tex]\[
\sqrt{36 x^6} = \sqrt{36} \cdot \sqrt{x^6}
\][/tex]
5. Simplify each part individually:
- The square root of 36 is 6:
[tex]\[
\sqrt{36} = 6
\][/tex]
- The square root of [tex]\(x^6\)[/tex] can be written as:
[tex]\[
\sqrt{x^6} = \sqrt{(x^3)^2} = x^3
\][/tex]
6. Combine the simplified parts:
[tex]\[
\sqrt{36 x^6} = 6 \cdot x^3
\][/tex]
Therefore, the simplified product is [tex]\(6 x^3\)[/tex]. There seems to be a slight confusion in the number options provided. Given the options, it looks like there is a mistake in options listed. The correct answer, as we simplified, is [tex]\(6 x^3\)[/tex].
However, since an assumption about the format of one of the choices matching our answer did not match, it is essential to check provided options as such or report rectification needed:
- [tex]\(\sqrt{6 x^4}\)[/tex]
- [tex]\(\sqrt{36 x^8}\)[/tex]
- [tex]\(18 x^4\)[/tex]
- [tex]\(6 x^4\)[/tex]
None correctly represents the simplified form as we calculated [tex]\(6x^3\)[/tex].
