To solve the problem of rationalizing the denominator for the expression [tex]\(\frac{\sqrt{3}}{\sqrt{8}}\)[/tex], let's go through the following steps:
1. Original Expression:
[tex]\[
\frac{\sqrt{3}}{\sqrt{8}}
\][/tex]
2. Rationalize the Denominator: We do this by eliminating any square roots in the denominator. To do this, multiply both the numerator and the denominator by [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
\frac{\sqrt{3}}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{\sqrt{3} \cdot \sqrt{8}}{\sqrt{8} \cdot \sqrt{8}}
\][/tex]
3. Simplify the Denominator:
[tex]\[
\sqrt{8} \cdot \sqrt{8} = 8
\][/tex]
So, the denominator becomes 8.
4. Simplify the Numerator:
[tex]\[
\sqrt{3} \cdot \sqrt{8} = \sqrt{3 \cdot 8} = \sqrt{24}
\][/tex]
5. Expression after Rationalizing:
[tex]\[
\frac{\sqrt{24}}{8}
\][/tex]
6. Further Simplify the Numerator: Simplify [tex]\(\sqrt{24}\)[/tex]. Recall that:
[tex]\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2 \sqrt{6}
\][/tex]
So now the expression becomes:
[tex]\[
\frac{2 \sqrt{6}}{8}
\][/tex]
7. Simplify the Fraction:
[tex]\[
\frac{2 \sqrt{6}}{8} = \frac{2}{8} \cdot \sqrt{6} = \frac{1}{4} \cdot \sqrt{6} = \frac{\sqrt{6}}{4}
\][/tex]
Therefore, the rationalized form of [tex]\(\frac{\sqrt{3}}{\sqrt{8}}\)[/tex] is [tex]\(\frac{\sqrt{6}}{4}\)[/tex].
The correct answer is:
[tex]\[
\boxed{\frac{\sqrt{6}}{4}}
\][/tex]
