To determine which of the given options is equivalent to the expression [tex]\(\frac{\sqrt{22}}{2 \sqrt{2}}\)[/tex], let's simplify the expression step-by-step.
1. Express the given fraction:
[tex]\[
\frac{\sqrt{22}}{2 \sqrt{2}}
\][/tex]
2. Rationalize the denominator:
To simplify the fraction, we can start by simplifying the square roots in the denominator. Note that [tex]\(\sqrt{2}\)[/tex] is already in its simplest form, but we can simplify the overall expression without needing to rationalize:
3. Combine the square roots in the numerator:
We can express [tex]\(\sqrt{22}\)[/tex] as [tex]\(\sqrt{22} = \sqrt{2 \cdot 11} = \sqrt{2} \cdot \sqrt{11}\)[/tex]. Substituting this into the original fraction gives:
[tex]\[
\frac{\sqrt{22}}{2 \sqrt{2}} = \frac{\sqrt{2} \cdot \sqrt{11}}{2 \sqrt{2}}
\][/tex]
4. Simplify the fraction:
The [tex]\(\sqrt{2}\)[/tex] terms in the numerator and the denominator can cancel out:
[tex]\[
\frac{\sqrt{2} \cdot \sqrt{11}}{2 \sqrt{2}} = \frac{\sqrt{11}}{2}
\][/tex]
So, the expression simplifies to:
[tex]\[
\frac{\sqrt{11}}{2}
\][/tex]
Finally, we compare the result with the given options:
- A. [tex]\(\sqrt{11}\)[/tex]
- B. [tex]\(\frac{\sqrt{11}}{2}\)[/tex]
- C. [tex]\(\frac{\sqrt{11}}{4}\)[/tex]
- D. [tex]\(2 \sqrt{11}\)[/tex]
The simplified form matches option B:
[tex]\(\frac{\sqrt{11}}{2}\)[/tex]
Therefore, the equivalent expression is:
[tex]\(\boxed{\frac{\sqrt{11}}{2}}\)[/tex]
