To determine which of the given expressions matches the quotient [tex]\(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\)[/tex], let's evaluate the numerical value of each option.
First, let's denote the numerator and the denominator of the original expression:
1. Numerator: [tex]\(\sqrt{6} + \sqrt{11}\)[/tex]
2. Denominator: [tex]\(\sqrt{5} + \sqrt{3}\)[/tex]
Evaluating the quotient, we find:
[tex]\[
\frac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}} \approx 1.4531
\][/tex]
Now, we will consider each of the given options:
1. [tex]\(\frac{\sqrt{30} + 3 \sqrt{2} + \sqrt{55} + \sqrt{33}}{8}\)[/tex]
Evaluate the numerator:
[tex]\[
\sqrt{30} + 3\sqrt{2} + \sqrt{55} + \sqrt{33} \approx 5.4772 + 4.2426 + 7.4162 + 5.7446 \approx 22.8806
\][/tex]
Then, the overall expression is:
[tex]\[
\frac{22.8806}{8} \approx 2.8601
\][/tex]
2. [tex]\(\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{55} - \sqrt{33}}{2}\)[/tex]
Evaluate the numerator:
[tex]\[
\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33} \approx 5.4772 - 4.2426 + 7.4162 - 5.7446 \approx 2.9062
\][/tex]
Then, the overall expression is:
[tex]\[
\frac{2.9062}{2} \approx 1.4531
\][/tex]
3. [tex]\(\frac{17}{8}\)[/tex]
Evaluate the expression directly:
[tex]\[
\frac{17}{8} = 2.125
\][/tex]
4. [tex]\(-\frac{5}{2}\)[/tex]
Evaluate the expression directly:
[tex]\[
-\frac{5}{2} = -2.5
\][/tex]
Comparing these values:
- Original quotient: [tex]\(1.4531\)[/tex]
- Option 1: [tex]\(2.8601\)[/tex]
- Option 2: [tex]\(1.4531\)[/tex]
- Option 3: [tex]\(2.125\)[/tex]
- Option 4: [tex]\(-2.5\)[/tex]
The correct choice that matches the value of the quotient [tex]\(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\)[/tex] is:
[tex]\[
\boxed{\frac{\sqrt{30} - 3 \sqrt{2} + \sqrt{55} - \sqrt{33}}{2}}
\][/tex]
