If x=5-2√6, then the value of `x^(2)+(1)/(x^(2))`is equal to :

Answer

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

### Step 1: Calculate \( x^2 \)

First, we need to square \( x \):
\[ x^2 = (5 - 2\sqrt{6})^2 \]
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ x^2 = 5^2 - 2 \cdot 5 \cdot 2\sqrt{6} + (2\sqrt{6})^2 \]
Calculating each term:
\[ = 25 - 20\sqrt{6} + 4 \cdot 6 \]
\[ = 25 - 20\sqrt{6} + 24 \]
\[ = 49 - 20\sqrt{6} \]

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

Next, we find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{5 - 2\sqrt{6}} \]
To rationalize the denominator, multiply the numerator and the denominator by the conjugate:
\[ \frac{1}{x} = \frac{5 + 2\sqrt{6}}{(5 - 2\sqrt{6})(5 + 2\sqrt{6})} \]
Calculating the denominator:
\[ (5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]
Thus, we have:
\[ \frac{1}{x} = 5 + 2\sqrt{6} \]

### Step 3: Calculate \( \frac{1}{x^2} \)

Now we square \( \frac{1}{x} \):
\[ \frac{1}{x^2} = (5 + 2\sqrt{6})^2 \]
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ = 5^2 + 2 \cdot 5 \cdot 2\sqrt{6} + (2\sqrt{6})^2 \]
Calculating each term:
\[ = 25 + 20\sqrt{6} + 24 \]
\[ = 49 + 20\sqrt{6} \]

### Step 4: Calculate \( x^2 + \frac{1}{x^2} \)

Now we can find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = (49 - 20\sqrt{6}) + (49 + 20\sqrt{6}) \]
The \( -20\sqrt{6} \) and \( +20\sqrt{6} \) cancel out:
\[ = 49 + 49 = 98 \]

### Final Answer

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