To simplify the expression [tex]\(7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}}\)[/tex], we follow these steps:
1. Simplify inside the square roots:
- [tex]\(\sqrt{2 x^4}\)[/tex]:
- [tex]\(\sqrt{2 x^4} = \sqrt{2} \cdot \sqrt{x^4} = \sqrt{2} \cdot x^2\)[/tex]
- [tex]\(\sqrt{2 x^{12}}\)[/tex]:
- [tex]\(\sqrt{2 x^{12}} = \sqrt{2} \cdot \sqrt{x^{12}} = \sqrt{2} \cdot x^6\)[/tex]
2. Substitute back into the given expression:
- [tex]\(7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}}\)[/tex]:
- [tex]\(7 x^2 (\sqrt{2} \cdot x^2) \cdot 6 (\sqrt{2} \cdot x^6)\)[/tex]
- [tex]\(7 x^2 (\sqrt{2} x^2) \cdot 6 (\sqrt{2} x^6)\)[/tex]
3. Combine and rearrange the constants, square roots, and like terms:
- Combine the constants:
- [tex]\(7 \cdot 6 = 42\)[/tex]
- Combine the square root terms:
- [tex]\(\sqrt{2} \cdot \sqrt{2} = 2\)[/tex]
- Combine the powers of [tex]\(x\)[/tex]:
- [tex]\(x^2 \cdot x^2 \cdot x^6 = x^{2+2+6} = x^{10}\)[/tex]
4. Put it all together:
- The expression becomes:
- [tex]\(42 \cdot 2 \cdot x^{10} = 84 x^{10}\)[/tex]
Thus, the equivalent expression is [tex]\(84 x^{10}\)[/tex].
The correct answer is:
[tex]\[ \boxed{84 x^{10}} \][/tex]
