The square root of ` ( (sqrt3 + sqrt2)/(sqrt3 - sqrt2))` is

Options

A. `sqrt3 + sqrt2`
B. `sqrt3 - sqrt2`
C. `sqrt3 pm sqrt3`
D. `sqrt2 - sqrt3`

Answer

To solve the problem of finding the square root of \(\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}\), we will follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \sqrt{\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}} \]

### Step 2: Rationalize the denominator
To simplify the expression, we will rationalize the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3} + \sqrt{2}\):
\[ \frac{(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]

### Step 3: Simplify the denominator
Using the difference of squares formula \(a^2 - b^2\), we can simplify the denominator:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]

### Step 4: Simplify the numerator
Now, we simplify the numerator:
\[ (\sqrt{3} + \sqrt{2})^2 = (\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{2}) + (\sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \]

### Step 5: Combine the results
Now, we can write the entire expression as:
\[ \frac{5 + 2\sqrt{6}}{1} = 5 + 2\sqrt{6} \]

### Step 6: Take the square root
Finally, we take the square root of the simplified expression:
\[ \sqrt{5 + 2\sqrt{6}} \]

### Final Answer
Thus, the square root of \(\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}\) is:
\[ \sqrt{5 + 2\sqrt{6}} \]