To find which choice is equivalent to the quotient [tex]\(\sqrt{7 x^2} \div \sqrt{3 x}\)[/tex] for acceptable values of [tex]\(x\)[/tex], let's simplify the given expression step-by-step.
1. Start with the given expression:
[tex]\[
\frac{\sqrt{7 x^2}}{\sqrt{3 x}}
\][/tex]
2. Combine the square roots into a single square root:
[tex]\[
\sqrt{\frac{7 x^2}{3 x}}
\][/tex]
3. Simplify the fraction inside the square root:
[tex]\[
\frac{7 x^2}{3 x} = \frac{7}{3} \cdot \frac{x^2}{x} = \frac{7}{3} \cdot x = \frac{7 x}{3}
\][/tex]
4. The simplified expression inside the square root is:
[tex]\[
\sqrt{\frac{7 x}{3}}
\][/tex]
Therefore, the expression [tex]\(\sqrt{7 x^2} \div \sqrt{3 x}\)[/tex] simplifies to [tex]\(\sqrt{\frac{7 x}{3}}\)[/tex].
So, the correct choice is:
[tex]\[
\boxed{\sqrt{\frac{7 x}{3}}}
\][/tex]
