Alright, let's work through this problem step-by-step.
First, we are given the value of [tex]\( x \)[/tex], which is:
[tex]\[ x = 11 - 2\sqrt{30} \][/tex]
### Step 1: Compute [tex]\( \sqrt{x} \)[/tex]
The first step is to find the square root of [tex]\( x \)[/tex]:
[tex]\[ \sqrt{x} = \sqrt{11 - 2\sqrt{30}} \][/tex]
Given the result from the calculations, we know:
[tex]\[ \sqrt{x} = 0.21342176528338816 \][/tex]
### Step 2: Compute the reciprocal of [tex]\( \sqrt{x} \)[/tex]
Next, we need to find the reciprocal of [tex]\( \sqrt{x} \)[/tex]:
[tex]\[ \frac{1}{\sqrt{x}} \][/tex]
Given the result from the calculations, we know:
[tex]\[ \frac{1}{\sqrt{x}} = 4.685557720282973 \][/tex]
### Step 3: Compute [tex]\( \sqrt{x} + \frac{1}{\sqrt{x}} \)[/tex]
Now, we will determine the value of the expression:
[tex]\[ \sqrt{x} + \frac{1}{\sqrt{x}} \][/tex]
Substitute the values we have:
[tex]\[ \sqrt{x} + \frac{1}{\sqrt{x}} = 0.21342176528338816 + 4.685557720282973 \][/tex]
Add these values together:
[tex]\[ \sqrt{x} + \frac{1}{\sqrt{x}} = 4.898979485566361 \][/tex]
### Step 4: Compute [tex]\( \sqrt{x} - \frac{1}{\sqrt{x}} \)[/tex]
Next, we determine the value of the expression:
[tex]\[ \sqrt{x} - \frac{1}{\sqrt{x}} \][/tex]
Substitute the values we have:
[tex]\[ \sqrt{x} - \frac{1}{\sqrt{x}} = 0.21342176528338816 - 4.685557720282973 \][/tex]
Subtract these values:
[tex]\[ \sqrt{x} - \frac{1}{\sqrt{x}} = -4.472135954999585 \][/tex]
### Summary
Thus, the values are:
(i) [tex]\(\sqrt{x} + \frac{1}{\sqrt{x}} = 4.898979485566361\)[/tex]
(ii) [tex]\(\sqrt{x} - \frac{1}{\sqrt{x}} = -4.472135954999585\)[/tex]
These are the required values for the given expressions.
