To find the expression equivalent to \(7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}}\), let's go through the steps of simplification step by step.
First, we need to simplify the square roots within the expression.
1. Simplify \(\sqrt{2 x^4}\):
[tex]\[
\sqrt{2 x^4} = \sqrt{2} \cdot \sqrt{x^4} = \sqrt{2} \cdot x^2
\][/tex]
2. Simplify \(\sqrt{2 x^{12}}\):
[tex]\[
\sqrt{2 x^{12}} = \sqrt{2} \cdot \sqrt{x^{12}} = \sqrt{2} \cdot x^6
\][/tex]
Now, substitute these simplified forms back into the original expression:
[tex]\[
7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}} = 7 x^2 (\sqrt{2} \cdot x^2) \cdot 6 (\sqrt{2} \cdot x^6)
\][/tex]
Next, combine like terms and constants:
[tex]\[
7 x^2 \cdot \sqrt{2} \cdot x^2 \cdot 6 \cdot \sqrt{2} \cdot x^6
\][/tex]
First, multiply the constants:
[tex]\[
7 \cdot 6 = 42
\][/tex]
Combine the square roots:
[tex]\[
\sqrt{2} \cdot \sqrt{2} = \sqrt{2^2} = \sqrt{4} = 2
\][/tex]
So the expression becomes:
[tex]\[
42 x^2 \cdot x^2 \cdot x^6 \cdot 2
\][/tex]
Combine the terms with \(x\):
[tex]\[
x^2 \cdot x^2 \cdot x^6 = x^{2+2+6} = x^{10}
\][/tex]
Now, multiply all components together:
[tex]\[
42 \cdot 2 \cdot x^{10} = 84 x^{10}
\][/tex]
Therefore, the equivalent expression is:
\[
\boxed{84 x^{10}}
\
