To solve the given expression [tex]\(\sqrt{\frac{1}{x^2}} \cdot \sqrt{\frac{x^2}{81}}\)[/tex] and determine which choice is equivalent to it when [tex]\(x > 0\)[/tex], follow these steps:
1. Simplify Each Square Root Separately:
[tex]\[
\sqrt{\frac{1}{x^2}} = \frac{1}{x}
\][/tex]
This is because the square root of [tex]\(\frac{1}{x^2}\)[/tex] is [tex]\(\frac{1}{x}\)[/tex].
[tex]\[
\sqrt{\frac{x^2}{81}} = \frac{x}{9}
\][/tex]
This is because the square root of [tex]\(\frac{x^2}{81}\)[/tex] is [tex]\(\frac{x}{9}\)[/tex].
2. Multiply the Simplified Expressions:
Now, we need to multiply the two simplified expressions together:
[tex]\[
\frac{1}{x} \cdot \frac{x}{9}
\][/tex]
3. Simplify the Product:
When multiplying, the [tex]\(x\)[/tex] in the numerator of the second fraction cancels out with the [tex]\(x\)[/tex] in the denominator of the first fraction:
[tex]\[
\frac{1 \cdot x}{x \cdot 9} = \frac{x}{9x} = \frac{1}{9}
\][/tex]
Therefore, the product [tex]\(\sqrt{\frac{1}{x^2}} \cdot \sqrt{\frac{x^2}{81}}\)[/tex] simplifies to [tex]\(\frac{1}{9}\)[/tex].
Hence, the correct choice is:
B. [tex]\(\frac{1}{9}\)[/tex]
