Let's solve the given question step by step, focusing on simplifying each term and then rewriting it in its simplified form.
### Simplification of Terms
1. Term: [tex]\(\sqrt{2x^3}\)[/tex]
- This term can be simplified as follows:
[tex]\[
\sqrt{2x^3} = \sqrt{2} \cdot \sqrt{x^3} = \sqrt{2} \cdot x^{3/2}
\][/tex]
2. Term: [tex]\(\sqrt{18x^5}\)[/tex]
- This term can be simplified as follows:
[tex]\[
\sqrt{18x^5} = \sqrt{18} \cdot \sqrt{x^5} = \sqrt{18} \cdot x^{5/2} = 3\sqrt{2} \cdot x^{5/2} = 3\sqrt{2} \cdot \sqrt{x^5}
\][/tex]
3. Term: [tex]\(\sqrt{6x^4}\)[/tex]
- This term can be simplified as follows:
[tex]\[
\sqrt{6x^4} = \sqrt{6} \cdot \sqrt{x^4} = \sqrt{6} \cdot x^2
\][/tex]
4. Term: [tex]\(\sqrt{36x^8}\)[/tex]
- This term can be simplified as follows:
[tex]\[
\sqrt{36x^8} = \sqrt{36} \cdot \sqrt{x^8} = 6 \cdot x^4
\][/tex]
### Given Simplified Values
- We also have two additional simplified terms:
- [tex]\(18x^4\)[/tex]
- [tex]\(6x^4\)[/tex]
### Results
Now, let's rewrite all the simplified terms together:
1. [tex]\(\sqrt{2x^3} = \sqrt{2} \cdot \sqrt{x^3}\)[/tex]
2. [tex]\(\sqrt{18x^5} = 3\sqrt{2} \cdot \sqrt{x^5}\)[/tex]
3. [tex]\(\sqrt{6x^4} = \sqrt{6} \cdot \sqrt{x^4}\)[/tex]
4. [tex]\(\sqrt{36x^8} = 6 \cdot \sqrt{x^8}\)[/tex]
5. [tex]\(18x^4\)[/tex]
6. [tex]\(6x^4\)[/tex]
Therefore, the simplified forms for the given expressions are:
- [tex]\(\sqrt{2} \cdot \sqrt{x^3}\)[/tex]
- [tex]\(3\sqrt{2} \cdot \sqrt{x^5}\)[/tex]
- [tex]\(\sqrt{6} \cdot \sqrt{x^4}\)[/tex]
- [tex]\(6 \cdot \sqrt{x^8}\)[/tex]
- [tex]\(18x^4\)[/tex]
- [tex]\(6x^4\)[/tex]
These are the simplified products of the given terms.
