Answer :
To determine which choice is equivalent to the given expression [tex]\(\sqrt{\frac{6}{x}} \cdot \sqrt{\frac{x^2}{24}}\)[/tex] when [tex]\(x > 0\)[/tex], we will simplify the expression step-by-step.
First, let's rewrite the given expression as a single square root:
[tex]\[ \sqrt{\frac{6}{x}} \cdot \sqrt{\frac{x^2}{24}} = \sqrt{\left( \frac{6}{x} \right) \cdot \left( \frac{x^2}{24} \right)} \][/tex]
Next, we combine the fractions under a single square root:
[tex]\[ \sqrt{\frac{6 \cdot x^2}{x \cdot 24}} = \sqrt{\frac{6x^2}{24x}} \][/tex]
We can simplify the fraction inside the square root:
[tex]\[ \sqrt{\frac{6x^2}{24x}} = \sqrt{\frac{6x}{24}} = \sqrt{\frac{x}{4}} \][/tex]
Further simplifying the square root:
[tex]\[ \sqrt{\frac{x}{4}} = \frac{\sqrt{x}}{\sqrt{4}} = \frac{\sqrt{x}}{2} \][/tex]
We have now simplified the given expression to:
[tex]\[ \sqrt{\frac{6}{x}} \cdot \sqrt{\frac{x^2}{24}} = \frac{\sqrt{x}}{2} \][/tex]
Therefore, the choice that is equivalent to the given expression is:
[tex]\[ \boxed{D} \][/tex]
First, let's rewrite the given expression as a single square root:
[tex]\[ \sqrt{\frac{6}{x}} \cdot \sqrt{\frac{x^2}{24}} = \sqrt{\left( \frac{6}{x} \right) \cdot \left( \frac{x^2}{24} \right)} \][/tex]
Next, we combine the fractions under a single square root:
[tex]\[ \sqrt{\frac{6 \cdot x^2}{x \cdot 24}} = \sqrt{\frac{6x^2}{24x}} \][/tex]
We can simplify the fraction inside the square root:
[tex]\[ \sqrt{\frac{6x^2}{24x}} = \sqrt{\frac{6x}{24}} = \sqrt{\frac{x}{4}} \][/tex]
Further simplifying the square root:
[tex]\[ \sqrt{\frac{x}{4}} = \frac{\sqrt{x}}{\sqrt{4}} = \frac{\sqrt{x}}{2} \][/tex]
We have now simplified the given expression to:
[tex]\[ \sqrt{\frac{6}{x}} \cdot \sqrt{\frac{x^2}{24}} = \frac{\sqrt{x}}{2} \][/tex]
Therefore, the choice that is equivalent to the given expression is:
[tex]\[ \boxed{D} \][/tex]
