To simplify the expression [tex]\(7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}}\)[/tex], we need to break down each part and simplify step-by-step.
1. Simplify [tex]\( \sqrt{2 x^4} \)[/tex]:
[tex]\[
\sqrt{2 x^4} = \sqrt{2} \cdot \sqrt{x^4} = \sqrt{2} \cdot x^2
\][/tex]
2. Substitute this back into the expression:
[tex]\[
7 x^2 \sqrt{2 x^4} = 7 x^2 \cdot (\sqrt{2} \cdot x^2) = 7 \sqrt{2} \cdot x^4
\][/tex]
3. Simplify [tex]\( \sqrt{2 x^{12}} \)[/tex]:
[tex]\[
\sqrt{2 x^{12}} = \sqrt{2} \cdot \sqrt{x^{12}} = \sqrt{2} \cdot x^6
\][/tex]
4. Substitute this back into the expression:
[tex]\[
6 \sqrt{2 x^{12}} = 6 \cdot (\sqrt{2} \cdot x^6) = 6 \sqrt{2} \cdot x^6
\][/tex]
5. Multiply the simplified terms together:
[tex]\[
(7 \sqrt{2} \cdot x^4) \cdot (6 \sqrt{2} \cdot x^6)
\][/tex]
6. Combine the coefficients and the square roots:
[tex]\[
7 \cdot 6 \cdot (\sqrt{2} \cdot \sqrt{2}) \cdot x^{4+6}
\][/tex]
[tex]\[
= 42 \cdot 2 \cdot x^{10}
\][/tex]
[tex]\[
= 84 x^{10}
\][/tex]
Therefore, the expression [tex]\(7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}}\)[/tex] simplifies to [tex]\(84 x^{10}\)[/tex].
The correct answer is:
C. [tex]\(84 x^{10}\)[/tex]
