Certainly! Let's go through the steps to find the value of [tex]\(\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)\)[/tex] when [tex]\(x = 5 + 2\sqrt{6}\)[/tex].
1. Calculate [tex]\(x\)[/tex]:
Given [tex]\(x = 5 + 2 \sqrt{6}\)[/tex].
2. Find [tex]\(\sqrt{x}\)[/tex]:
Calculate [tex]\(\sqrt{5 + 2 \sqrt{6}}\)[/tex].
After calculating, we find:
[tex]\[
\sqrt{5 + 2 \sqrt{6}} = 3.146264369941972
\][/tex]
3. Find [tex]\(\frac{1}{\sqrt{x}}\)[/tex]:
Use the value found for [tex]\(\sqrt{x}\)[/tex] to calculate [tex]\(\frac{1}{\sqrt{x}}\)[/tex]:
[tex]\[
\frac{1}{\sqrt{x}} = \frac{1}{3.146264369941972} = 0.31783724519578227
\][/tex]
4. Calculate [tex]\(\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)\)[/tex]:
Subtract [tex]\(\frac{1}{\sqrt{x}}\)[/tex] from [tex]\(\sqrt{x}\)[/tex]:
[tex]\[
\sqrt{x} - \frac{1}{\sqrt{x}} = 3.146264369941972 - 0.31783724519578227 = 2.82842712474619
\][/tex]
Therefore, the value of [tex]\(\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)\)[/tex] is:
[tex]\[
\boxed{2.82842712474619}
\][/tex]
Since this numerical result matches [tex]\(\mathbf{2 \sqrt{2}}\)[/tex], our answer corresponds to option B:
[tex]\[
\boxed{2 \sqrt{2}}
\][/tex]
