If `x=2sqrt(3)+2sqrt(2)`, find :
(i) `1/x" (ii) "x+1/x" (iii) "(x+1/x)^(2)`

Answer

To solve the problem step by step, we will follow the instructions given in the question.

Given:
\[ x = 2\sqrt{3} + 2\sqrt{2} \]

We need to find:
(i) \( \frac{1}{x} \)
(ii) \( x + \frac{1}{x} \)
(iii) \( \left( x + \frac{1}{x} \right)^2 \)

### Step 1: Find \( \frac{1}{x} \)

To find \( \frac{1}{x} \), we can rationalize the denominator.

1. Start with:
\[ \frac{1}{x} = \frac{1}{2\sqrt{3} + 2\sqrt{2}} \]

2. Factor out \( 2 \) from the denominator:
\[ \frac{1}{x} = \frac{1}{2(\sqrt{3} + \sqrt{2})} = \frac{1}{2} \cdot \frac{1}{\sqrt{3} + \sqrt{2}} \]

3. Rationalize the denominator by multiplying the numerator and denominator by \( \sqrt{3} - \sqrt{2} \):
\[ \frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2} \]

4. Therefore:
\[ \frac{1}{x} = \frac{1}{2}(\sqrt{3} - \sqrt{2}) \]

### Step 2: Find \( x + \frac{1}{x} \)

Now, we can find \( x + \frac{1}{x} \):
1. Substitute the values:
\[ x + \frac{1}{x} = (2\sqrt{3} + 2\sqrt{2}) + \frac{1}{2}(\sqrt{3} - \sqrt{2}) \]

2. Combine the terms:
\[ = 2\sqrt{3} + 2\sqrt{2} + \frac{1}{2}\sqrt{3} - \frac{1}{2}\sqrt{2} \]
\[ = \left(2 + \frac{1}{2}\right)\sqrt{3} + \left(2 - \frac{1}{2}\right)\sqrt{2} \]
\[ = \frac{5}{2}\sqrt{3} + \frac{3}{2}\sqrt{2} \]

### Step 3: Find \( \left( x + \frac{1}{x} \right)^2 \)

Now we will square the result from Step 2:
1. Use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ \left( \frac{5}{2}\sqrt{3} + \frac{3}{2}\sqrt{2} \right)^2 \]
\[ = \left(\frac{5}{2}\sqrt{3}\right)^2 + 2\left(\frac{5}{2}\sqrt{3}\right)\left(\frac{3}{2}\sqrt{2}\right) + \left(\frac{3}{2}\sqrt{2}\right)^2 \]

2. Calculate each term:
- First term:
\[ \left(\frac{5}{2}\sqrt{3}\right)^2 = \frac{25}{4} \cdot 3 = \frac{75}{4} \]
- Second term:
\[ 2 \cdot \frac{5}{2}\sqrt{3} \cdot \frac{3}{2}\sqrt{2} = \frac{15}{2}\sqrt{6} \]
- Third term:
\[ \left(\frac{3}{2}\sqrt{2}\right)^2 = \frac{9}{4} \cdot 2 = \frac{18}{4} = \frac{9}{2} \]

3. Combine all terms:
\[ \left( x + \frac{1}{x} \right)^2 = \frac{75}{4} + \frac{15}{2}\sqrt{6} + \frac{9}{2} \]
\[ = \frac{75}{4} + \frac{18}{4} + \frac{15}{2}\sqrt{6} \]
\[ = \frac{93}{4} + \frac{30}{4}\sqrt{6} \]

### Final Answers:
(i) \( \frac{1}{x} = \frac{\sqrt{3} - \sqrt{2}}{2} \)
(ii) \( x + \frac{1}{x} = \frac{5}{2}\sqrt{3} + \frac{3}{2}\sqrt{2} \)
(iii) \( \left( x + \frac{1}{x} \right)^2 = \frac{93}{4} + \frac{30}{4}\sqrt{6} \)