To determine the equivalent expression for [tex]\( \frac{18 x^2 \sqrt{14 x^8}}{6 \sqrt{7 x^4}} \)[/tex] where [tex]\( x \neq 0 \)[/tex], we need to simplify step-by-step. Let's go through the process thoroughly:
1. Simplify the Radicals:
- First, look at the term [tex]\( \sqrt{14 x^8} \)[/tex]:
[tex]\[
\sqrt{14 x^8} = \sqrt{14} \cdot \sqrt{x^8} = \sqrt{14} \cdot x^4
\][/tex]
- Next, look at the term [tex]\( \sqrt{7 x^4} \)[/tex]:
[tex]\[
\sqrt{7 x^4} = \sqrt{7} \cdot \sqrt{x^4} = \sqrt{7} \cdot x^2
\][/tex]
2. Rewrite the Expression:
Substitute the simplified radicals back into the original expression:
[tex]\[
\frac{18 x^2 \left( \sqrt{14} \cdot x^4 \right)}{6 \left( \sqrt{7} \cdot x^2 \right)}
\][/tex]
3. Simplify Inside the Fraction:
Combine the [tex]\( x \)[/tex] terms and the coefficients:
[tex]\[
\frac{18 x^2 \cdot \sqrt{14} \cdot x^4}{6 \cdot \sqrt{7} \cdot x^2} = \frac{18 \cdot \sqrt{14} \cdot x^6}{6 \cdot \sqrt{7} \cdot x^2}
\][/tex]
4. Cancel Out Common Terms:
- Simplify the coefficients:
[tex]\[
\frac{18}{6} = 3
\][/tex]
- Simplify the radicals:
[tex]\[
\frac{\sqrt{14}}{\sqrt{7}} = \sqrt{\frac{14}{7}} = \sqrt{2}
\][/tex]
- Simplify the [tex]\( x \)[/tex] terms:
[tex]\[
\frac{x^6}{x^2} = x^{6-2} = x^4
\][/tex]
5. Combine the Simplified Parts:
Putting it all together:
[tex]\[
3 \cdot \sqrt{2} \cdot x^4 = 3 x^4 \sqrt{2}
\][/tex]
Thus, the expression equivalent to [tex]\( \frac{18 x^2 \sqrt{14 x^8}}{6 \sqrt{7 x^4}} \)[/tex] is:
[tex]\[
\boxed{3 x^4 \sqrt{2}}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\text{B. } 3 x^4 \sqrt{2}
\][/tex]
