If x=`2sqrt(3)+2sqrt(2)` find :
`(x+(1)/(x))^(2)`

Answer

To solve the problem, we need to find the value of \((x + \frac{1}{x})^2\) where \(x = 2\sqrt{3} + 2\sqrt{2}\).

### Step-by-Step Solution:

1. **Calculate \(x + \frac{1}{x}\)**:
\[ x + \frac{1}{x} = (2\sqrt{3} + 2\sqrt{2}) + \frac{1}{(2\sqrt{3} + 2\sqrt{2})} \]

2. **Find \(\frac{1}{x}\)**:
To find \(\frac{1}{x}\), we will rationalize it:
\[ \frac{1}{x} = \frac{1}{2\sqrt{3} + 2\sqrt{2}} \cdot \frac{2\sqrt{3} - 2\sqrt{2}}{2\sqrt{3} - 2\sqrt{2}} = \frac{2\sqrt{3} - 2\sqrt{2}}{(2\sqrt{3})^2 - (2\sqrt{2})^2} \]
The denominator simplifies as follows:
\[ (2\sqrt{3})^2 - (2\sqrt{2})^2 = 4 \cdot 3 - 4 \cdot 2 = 12 - 8 = 4 \]
Thus,
\[ \frac{1}{x} = \frac{2\sqrt{3} - 2\sqrt{2}}{4} = \frac{\sqrt{3} - \sqrt{2}}{2} \]

3. **Combine \(x\) and \(\frac{1}{x}\)**:
Now, substituting back:
\[ x + \frac{1}{x} = (2\sqrt{3} + 2\sqrt{2}) + \left(\frac{\sqrt{3} - \sqrt{2}}{2}\right) \]
To combine these, we need a common denominator:
\[ = \frac{4\sqrt{3} + 4\sqrt{2}}{2} + \frac{\sqrt{3} - \sqrt{2}}{2} = \frac{(4\sqrt{3} + 4\sqrt{2} + \sqrt{3} - \sqrt{2})}{2} \]
Simplifying further:
\[ = \frac{(4\sqrt{3} + \sqrt{3}) + (4\sqrt{2} - \sqrt{2})}{2} = \frac{5\sqrt{3} + 3\sqrt{2}}{2} \]

4. **Square the result**:
Now we need to square \(x + \frac{1}{x}\):
\[ (x + \frac{1}{x})^2 = \left(\frac{5\sqrt{3} + 3\sqrt{2}}{2}\right)^2 \]
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[ = \frac{(5\sqrt{3})^2 + 2(5\sqrt{3})(3\sqrt{2}) + (3\sqrt{2})^2}{4} \]
Calculating each term:
\[ (5\sqrt{3})^2 = 25 \cdot 3 = 75 \]
\[ (3\sqrt{2})^2 = 9 \cdot 2 = 18 \]
\[ 2(5\sqrt{3})(3\sqrt{2}) = 30\sqrt{6} \]
Thus,
\[ = \frac{75 + 18 + 30\sqrt{6}}{4} = \frac{93 + 30\sqrt{6}}{4} \]

### Final Answer:
\[ (x + \frac{1}{x})^2 = \frac{93 + 30\sqrt{6}}{4} \]