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

Answer

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

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

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 2\sqrt{3} - 2\sqrt{2} \):
\[ \frac{1}{2\sqrt{3} + 2\sqrt{2}} \cdot \frac{2\sqrt{3} - 2\sqrt{2}}{2\sqrt{3} - 2\sqrt{2}} = \frac{2\sqrt{3} - 2\sqrt{2}}{(2\sqrt{3} + 2\sqrt{2})(2\sqrt{3} - 2\sqrt{2})} \]

### Step 3: Simplify the denominator using the difference of squares
Using the difference of squares formula \( (a+b)(a-b) = a^2 - b^2 \):
\[ (2\sqrt{3})^2 - (2\sqrt{2})^2 = 4 \cdot 3 - 4 \cdot 2 = 12 - 8 = 4 \]
So, the denominator simplifies to \( 4 \).

### Step 4: Write the simplified expression
Now we can write the expression for \( \frac{1}{x} \):
\[ \frac{2\sqrt{3} - 2\sqrt{2}}{4} \]

### Step 5: Simplify the fraction
We can simplify the fraction by dividing the numerator by 2:
\[ \frac{2\sqrt{3} - 2\sqrt{2}}{4} = \frac{\sqrt{3} - \sqrt{2}}{2} \]

### Final Result
Thus, the final result is:
\[ \frac{1}{x} = \frac{\sqrt{3} - \sqrt{2}}{2} \]

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