Which choice is equivalent to the quotient shown here for acceptable values of [tex]\(x\)[/tex]?

[tex]\[
\frac{\sqrt{7 x^2}}{\sqrt{3 x}}
\][/tex]

A. [tex]\(\sqrt{\frac{7 x}{3}}\)[/tex]

B. [tex]\(\sqrt{\frac{7 x^3}{3}}\)[/tex]

C. [tex]\(x \sqrt{\frac{7 x}{3}}\)[/tex]

D. [tex]\(\sqrt{21 x^3}\)[/tex]



Answer :

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]