To solve the expression [tex]\(\frac{b}{2 \sqrt{11}} + \frac{7 \sqrt{11} + 5 \sqrt{11}}{8}\)[/tex], we need to simplify each term separately and then combine them. Let's break it down step by step:
1. Simplify the first term:
[tex]\[
\frac{b}{2 \sqrt{11}}
\][/tex]
This term is already in its simplest form. We can leave it as is.
2. Simplify the second term:
First, note that [tex]\(7 \sqrt{11}\)[/tex] and [tex]\(5 \sqrt{11}\)[/tex] are like terms and can be added together:
[tex]\[
7 \sqrt{11} + 5 \sqrt{11} = (7 + 5) \sqrt{11} = 12 \sqrt{11}
\][/tex]
Now, substitute this back into the expression:
[tex]\[
\frac{12 \sqrt{11}}{8}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{12 \sqrt{11}}{8} = \frac{12}{8} \sqrt{11} = \frac{3}{2} \sqrt{11}
\][/tex]
3. Combine the simplified terms:
We now have the simplified form of each term:
[tex]\[
\frac{b}{2 \sqrt{11}} + \frac{3 \sqrt{11}}{2}
\][/tex]
To make the addition easier, let's get a common denominator. Notice that [tex]\(\frac{b}{2 \sqrt{11}}\)[/tex] can be rewritten in terms of [tex]\(\sqrt{11}\)[/tex]:
[tex]\[
\frac{b}{2 \sqrt{11}} = \frac{\sqrt{11} b}{22}
\][/tex]
The second term, [tex]\(\frac{3 \sqrt{11}}{2}\)[/tex], already has [tex]\(\sqrt{11}\)[/tex] in it:
[tex]\[
\frac{3 \sqrt{11}}{2}
\][/tex]
Now, add the two terms together:
[tex]\[
\frac{\sqrt{11} b}{22} + \frac{3 \sqrt{11}}{2}
\][/tex]
The result can't be simplified further without additional context, so the final answer is:
[tex]\[
\frac{\sqrt{11} b}{22} + \frac{3 \sqrt{11}}{2}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{\sqrt{11} b}{22} + \frac{3 \sqrt{11}}{2}}
\][/tex]
