If ` x= 5 - 2 sqrt6 , ` find the value of :
`x^(2) +(1)/(x^(2))`

Answer

To find the value of \( x^2 + \frac{1}{x^2} \) given \( x = 5 - 2\sqrt{6} \), we can follow these steps:

### Step 1: Find \( \frac{1}{x} \)
We start by calculating \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{5 - 2\sqrt{6}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{x} = \frac{1 \cdot (5 + 2\sqrt{6})}{(5 - 2\sqrt{6})(5 + 2\sqrt{6})} \]

### Step 2: Calculate the Denominator
Now, we calculate the denominator using the difference of squares:
\[ (5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]
Thus, we have:
\[ \frac{1}{x} = 5 + 2\sqrt{6} \]

### Step 3: Calculate \( x + \frac{1}{x} \)
Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (5 - 2\sqrt{6}) + (5 + 2\sqrt{6}) = 5 - 2\sqrt{6} + 5 + 2\sqrt{6} \]
The \( -2\sqrt{6} \) and \( +2\sqrt{6} \) cancel each other out:
\[ x + \frac{1}{x} = 10 \]

### Step 4: Square \( x + \frac{1}{x} \)
Next, we square both sides:
\[ \left( x + \frac{1}{x} \right)^2 = 10^2 \]
This gives us:
\[ x^2 + 2 + \frac{1}{x^2} = 100 \]

### Step 5: Solve for \( x^2 + \frac{1}{x^2} \)
Now, we can isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 100 - 2 = 98 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{98} \]