If `x= sqrt3 + sqrt2`, then value of `(x^(2) + (1)/(x^(2)))` is

Options

A. `2 sqrt3`
B. 10
C. 12
D. 14

Answer

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

### Step 1: Calculate \( x^2 \)
First, we calculate \( x^2 \):
\[ x^2 = (\sqrt{3} + \sqrt{2})^2 = (\sqrt{3})^2 + 2 \cdot \sqrt{3} \cdot \sqrt{2} + (\sqrt{2})^2 \]
\[ = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \]

### Step 2: Calculate \( \frac{1}{x} \)
Next, we need to find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{\sqrt{3} + \sqrt{2}} \]
To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt{3} - \sqrt{2} \):
\[ \frac{1}{x} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2} \]

### Step 3: Calculate \( \frac{1}{x^2} \)
Now we calculate \( \frac{1}{x^2} \):
\[ \frac{1}{x^2} = (\frac{1}{x})^2 = (\sqrt{3} - \sqrt{2})^2 = (\sqrt{3})^2 - 2 \cdot \sqrt{3} \cdot \sqrt{2} + (\sqrt{2})^2 \]
\[ = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6} \]

### Step 4: Calculate \( x^2 + \frac{1}{x^2} \)
Now we can find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 5 + 2\sqrt{6} + 5 - 2\sqrt{6} \]
\[ = 10 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is \( \boxed{10} \).