Question

which is equivalent to \\(\frac{\sqrt{3}}{\sqrt{8}}\\) when the denominator is
ationalized\?\
\\(\bigcirc\\) a. \\(\frac{3}{8}\\)\
\\(\bigcirc\\) b. \\(\frac{\sqrt{6}}{4}\\)\
\\(\bigcirc\\) c. \\(\frac{\sqrt{3}}{8}\\)\
\\(\bigcirc\\) d. \\(\frac{\sqrt{24}}{64}\\)\
\\(\bigcirc\\) e. \\(\frac{8\sqrt{3}}{3}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator of \(\frac{\sqrt{3}}{\sqrt{8}}\), we multiply the numerator and denominator by \(\sqrt{8}\) (or simplify \(\sqrt{8}\) first). First, simplify \(\sqrt{8}=\sqrt{4\times2} = 2\sqrt{2}\). But another way is to multiply numerator and denominator by \(\sqrt{8}\):
\[
\frac{\sqrt{3}}{\sqrt{8}}\times\frac{\sqrt{8}}{\sqrt{8}}=\frac{\sqrt{3\times8}}{8}=\frac{\sqrt{24}}{8}
\]
Simplify \(\sqrt{24}=\sqrt{4\times6}=2\sqrt{6}\), so \(\frac{2\sqrt{6}}{8}=\frac{\sqrt{6}}{4}\)? Wait, no, wait. Wait, \(\sqrt{8}\) simplified is \(2\sqrt{2}\), so let's do it again. Wait, maybe I made a mistake. Let's start over.

The expression is \(\frac{\sqrt{3}}{\sqrt{8}}\). We can also simplify \(\sqrt{8}\) as \(2\sqrt{2}\), so the expression becomes \(\frac{\sqrt{3}}{2\sqrt{2}}\). Now, to rationalize the denominator (get rid of the square root in the denominator), we multiply numerator and denominator by \(\sqrt{2}\):
\[
\frac{\sqrt{3}}{2\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{3\times2}}{2\times2}=\frac{\sqrt{6}}{4}
\]
Wait, but let's check the options. Option B is \(\frac{\sqrt{6}}{4}\)? Wait, no, the option B is \(\frac{\sqrt{6}}{4}\)? Wait, the option B is written as \(\frac{\sqrt{6}}{4}\)? Wait, the user's option B is \(\frac{\sqrt{6}}{4}\)? Wait, looking at the user's image, option B is \(\frac{\sqrt{6}}{4}\)? Wait, the user's option B: "B. \(\frac{\sqrt{6}}{4}\)"? Wait, the user's text: "B. \(\frac{\sqrt{6}}{4}\)"? Wait, the user's input: "B. \(\frac{\sqrt{6}}{4}\)"? Wait, in the image, the options are:

A. \(\frac{3}{8}\)

B. \(\frac{\sqrt{6}}{4}\)

C. \(\frac{\sqrt{3}}{8}\)

D. \(\frac{\sqrt{24}}{64}\)

E. \(\frac{8\sqrt{3}}{3}\)

Wait, maybe I misread. Let's do the correct rationalization.

\(\frac{\sqrt{3}}{\sqrt{8}}\). Multiply numerator and denominator by \(\sqrt{8}\):

\[
\frac{\sqrt{3}\times\sqrt{8}}{\sqrt{8}\times\sqrt{8}}=\frac{\sqrt{24}}{8}
\]

Simplify \(\sqrt{24}=\sqrt{4\times6}=2\sqrt{6}\), so \(\frac{2\sqrt{6}}{8}=\frac{\sqrt{6}}{4}\). Wait, but let's check the options. Option B is \(\frac{\sqrt{6}}{4}\)? Wait, the user's option B: "B. \(\frac{\sqrt{6}}{4}\)"? Yes. Wait, but let's verify again.

Alternatively, simplify \(\sqrt{8}\) as \(2\sqrt{2}\), so:

\[
\frac{\sqrt{3}}{2\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{6}}{4}
\]

Yes, that's correct. So the equivalent expression after rationalizing the denominator is \(\frac{\sqrt{6}}{4}\), which is option B.

Wait, but let's check the options again. The user's options:

A. \(\frac{3}{8}\)

B. \(\frac{\sqrt{6}}{4}\)

C. \(\frac{\sqrt{3}}{8}\)

D. \(\frac{\sqrt{24}}{64}\)

E. \(\frac{8\sqrt{3}}{3}\)

So the correct answer is B.

Answer:

B. \(\frac{\sqrt{6}}{4}\)