Sure, let's go through the given problem step by step.
We start with the given value:
[tex]\[ x = 5 + 2\sqrt{6} \][/tex]
The next step involves taking the square root of [tex]\( x \)[/tex]:
[tex]\[ \sqrt{x} = \sqrt{5 + 2\sqrt{6}} \][/tex]
Let's assume [tex]\(\sqrt{x}\)[/tex] is approximately equal to some value. According to the precise numerical result:
[tex]\[ \sqrt{x} \approx 3.146264369941972 \][/tex]
Now, we need to compute the reciprocal of [tex]\(\sqrt{x}\)[/tex] and calculate the following expression:
[tex]\[ \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right) \][/tex]
Given the value of [tex]\(\sqrt{x}\)[/tex] is approximately [tex]\( 3.146264369941972 \)[/tex], we find:
[tex]\[ \frac{1}{\sqrt{x}} \approx \frac{1}{3.146264369941972} \][/tex]
The numerical result already tells us that:
[tex]\[ \sqrt{x} - \frac{1}{\sqrt{x}} \approx 2.82842712474619 \][/tex]
Given the result:
[tex]\[ \sqrt{x} - \frac{1}{\sqrt{x}} \approx 2.82842712474619 \][/tex]
We can identify that this value corresponds to:
[tex]\[\sqrt{8} = 2\sqrt{2} \][/tex]
Thus, the value of the given expression is:
[tex]\[ \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right) \approx 2\sqrt{2} \][/tex]
Hence, the correct answer is:
[tex]\[ B. \, 2\sqrt{2} \][/tex]
