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Question 3:

If [tex]x = 5 + 2 \sqrt{6}[/tex], then find the value of [tex]\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)[/tex].

A. [tex]2 \sqrt{3}[/tex]

B. [tex]2 \sqrt{2}[/tex]

C. [tex]4 \sqrt{2}[/tex]

D. [tex]5 \sqrt{2}[/tex]

E. None of these



Answer :

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]