To find the value of the expression [tex]\(\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)\)[/tex] given [tex]\(x = 3 + 2\sqrt{2}\)[/tex], follow these detailed steps:
1. Substitute the value of [tex]\(x\)[/tex]:
Given [tex]\(x = 3 + 2\sqrt{2}\)[/tex].
2. Calculate the square root of [tex]\(x\)[/tex]:
[tex]\[
\sqrt{x} = \sqrt{3 + 2\sqrt{2}} \approx 2.414213562373095
\][/tex]
3. Find the reciprocal of the square root of [tex]\(x\)[/tex]:
[tex]\[
\frac{1}{\sqrt{x}} \approx \frac{1}{2.414213562373095} = 0.4142135623730951
\][/tex]
4. Determine the desired expression:
[tex]\[
\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right) = 2.414213562373095 - 0.4142135623730951 = 1.9999999999999998
\][/tex]
Therefore, the value of [tex]\(\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)\)[/tex] is approximately [tex]\(2\)[/tex].
