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

Answer

To solve the problem where \( x = 2\sqrt{3} + 2\sqrt{2} \) and we need to find \( x + \frac{1}{x} \), we can follow these steps:

### Step 1: Write down the expression
We start with the expression we need to evaluate:
\[ x + \frac{1}{x} = (2\sqrt{3} + 2\sqrt{2}) + \frac{1}{2\sqrt{3} + 2\sqrt{2}} \]

### Step 2: Simplify the expression
To simplify \( \frac{1}{x} \), we first need to rationalize the denominator:
\[ \frac{1}{2\sqrt{3} + 2\sqrt{2}} \cdot \frac{2\sqrt{3} - 2\sqrt{2}}{2\sqrt{3} - 2\sqrt{2}} = \frac{2\sqrt{3} - 2\sqrt{2}}{(2\sqrt{3})^2 - (2\sqrt{2})^2} \]

### Step 3: Calculate the denominator
Now, we calculate the denominator:
\[ (2\sqrt{3})^2 - (2\sqrt{2})^2 = 4 \cdot 3 - 4 \cdot 2 = 12 - 8 = 4 \]
So, we have:
\[ \frac{1}{2\sqrt{3} + 2\sqrt{2}} = \frac{2\sqrt{3} - 2\sqrt{2}}{4} = \frac{1}{2}(\sqrt{3} - \sqrt{2}) \]

### Step 4: Substitute back into the expression
Now, we substitute this back into our expression for \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (2\sqrt{3} + 2\sqrt{2}) + \frac{1}{2}(\sqrt{3} - \sqrt{2}) \]

### Step 5: Combine like terms
Now we 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 6: Final expression
Thus, the final result is:
\[ x + \frac{1}{x} = \frac{5}{2}\sqrt{3} + \frac{3}{2}\sqrt{2} \]