If `x = (sqrt3 + sqrt2)/(sqrt3 - sqrt2) ,` then the value of `(x + (1)/(x))` is :

Options

A. A)8
B. B)10
C. C)12
D. D)16

Answer

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

### Step 1: Simplify \( x \)

We start with the expression for \( x \):
\[ x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \]
To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{3} + \sqrt{2} \):
\[ x = \frac{(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]

### Step 2: Calculate the numerator and denominator

Calculating the numerator:
\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{3}\sqrt{2} = 5 + 2\sqrt{6} \]

Calculating the denominator:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]

So, we have:
\[ x = \frac{5 + 2\sqrt{6}}{1} = 5 + 2\sqrt{6} \]

### Step 3: Find \( \frac{1}{x} \)

Now, we need to find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{5 + 2\sqrt{6}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{x} = \frac{1 \cdot (5 - 2\sqrt{6})}{(5 + 2\sqrt{6})(5 - 2\sqrt{6})} \]

Calculating the denominator:
\[ (5)^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]

Thus, we have:
\[ \frac{1}{x} = 5 - 2\sqrt{6} \]

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

Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) \]
The \( 2\sqrt{6} \) terms cancel out:
\[ x + \frac{1}{x} = 5 + 5 = 10 \]

### Final Answer

Thus, the value of \( x + \frac{1}{x} \) is:
\[ \boxed{10} \]