Answer :
To solve the given problem [tex]\(\sqrt{\frac{5+2 \sqrt{6}}{5-2 \sqrt{6}}} - \sqrt{24}\)[/tex], we first simplify each part of the expression.
### Simplify [tex]\(\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}}\)[/tex]
First, simplify the fraction under the square root:
[tex]\[\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}}\][/tex]
### Simplify [tex]\(\sqrt{5 + 2 \sqrt{6}}\)[/tex] and [tex]\(\sqrt{5 - 2 \sqrt{6}}\)[/tex]
The expression [tex]\(\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}}\)[/tex] simplifies to:
[tex]\[\sqrt{2\sqrt{6} + 5} / \sqrt{5 - 2\sqrt{6}}\][/tex]
### Simplify [tex]\(\sqrt{24}\)[/tex]
Next, simplify [tex]\(\sqrt{24}\)[/tex]:
[tex]\[\sqrt{24}\][/tex]
This can be simplified as:
[tex]\[2 \sqrt{6}\][/tex]
### Simplify the overall expression
We now have two parts:
1. [tex]\(\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}}\)[/tex]
2. [tex]\(\sqrt{24}\)[/tex], which simplifies to [tex]\(2 \sqrt{6}\)[/tex]
We need to find:
[tex]\[\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}} - 2 \sqrt{6}\][/tex]
Combining these, we obtain the simplified form:
[tex]\[\frac{-2 \sqrt{30 - 12 \sqrt{6}} + \sqrt{2 \sqrt{6} + 5}}{\sqrt{5 - 2 \sqrt{6}}}\][/tex]
### Check the result against multiple-choice answers:
Given options:
A. 5
B. [tex]\(12 \sqrt{2}\)[/tex]
C. [tex]\(\frac{18 \sqrt{6}}{5}\)[/tex]
D. [tex]\(\frac{8 \sqrt{6}}{5}\)[/tex]
After simplification, the correct answer does not match any of the provided multiple-choice options. Therefore, the simplified value of [tex]\(\sqrt{\frac{5+2 \sqrt{6}}{5-2 \sqrt{6}}} - \sqrt{24}\)[/tex] is indeed none of the given choices.
Thus, the correct option is not listed because the correct value is not among the options [tex]\(A, B, C, D\)[/tex].
### Simplify [tex]\(\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}}\)[/tex]
First, simplify the fraction under the square root:
[tex]\[\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}}\][/tex]
### Simplify [tex]\(\sqrt{5 + 2 \sqrt{6}}\)[/tex] and [tex]\(\sqrt{5 - 2 \sqrt{6}}\)[/tex]
The expression [tex]\(\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}}\)[/tex] simplifies to:
[tex]\[\sqrt{2\sqrt{6} + 5} / \sqrt{5 - 2\sqrt{6}}\][/tex]
### Simplify [tex]\(\sqrt{24}\)[/tex]
Next, simplify [tex]\(\sqrt{24}\)[/tex]:
[tex]\[\sqrt{24}\][/tex]
This can be simplified as:
[tex]\[2 \sqrt{6}\][/tex]
### Simplify the overall expression
We now have two parts:
1. [tex]\(\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}}\)[/tex]
2. [tex]\(\sqrt{24}\)[/tex], which simplifies to [tex]\(2 \sqrt{6}\)[/tex]
We need to find:
[tex]\[\sqrt{\frac{5 + 2 \sqrt{6}}{5 - 2 \sqrt{6}}} - 2 \sqrt{6}\][/tex]
Combining these, we obtain the simplified form:
[tex]\[\frac{-2 \sqrt{30 - 12 \sqrt{6}} + \sqrt{2 \sqrt{6} + 5}}{\sqrt{5 - 2 \sqrt{6}}}\][/tex]
### Check the result against multiple-choice answers:
Given options:
A. 5
B. [tex]\(12 \sqrt{2}\)[/tex]
C. [tex]\(\frac{18 \sqrt{6}}{5}\)[/tex]
D. [tex]\(\frac{8 \sqrt{6}}{5}\)[/tex]
After simplification, the correct answer does not match any of the provided multiple-choice options. Therefore, the simplified value of [tex]\(\sqrt{\frac{5+2 \sqrt{6}}{5-2 \sqrt{6}}} - \sqrt{24}\)[/tex] is indeed none of the given choices.
Thus, the correct option is not listed because the correct value is not among the options [tex]\(A, B, C, D\)[/tex].
