Select the correct answer.

Which expression is equivalent to [tex]$18 x^2 \sqrt{14 x^8} \div 6 \sqrt{7 x^4}$[/tex], if [tex]$x \neq 0$[/tex]?

A. [tex]$12 x^4 \sqrt{2}$[/tex]
B. [tex][tex]$3 x^4 \sqrt{2}$[/tex][/tex]
C. [tex]$3 x^4 \sqrt{7}$[/tex]
D. [tex]$3 x \sqrt{2}$[/tex]



Answer :

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]