If `p=5+2sqrt(6)` then `(sqrt(p)-1)/(sqrt(p))` is

Options

A. `1+sqrt(2)-sqrt(3)`
B. `1-sqrt(2)+sqrt(3)`
C. `-1+sqrt(2)-sqrt(3)`
D. `1-sqrt(2)-sqrt(3)`

Answer

To solve the problem, we need to find the value of \(\frac{\sqrt{p} - 1}{\sqrt{p}}\) given that \(p = 5 + 2\sqrt{6}\).

### Step-by-Step Solution:

1. **Identify \(p\)**:
\[ p = 5 + 2\sqrt{6} \]

2. **Calculate \(\sqrt{p}\)**:
We can express \(p\) in a different form. Notice that:
\[ p = (\sqrt{3} + \sqrt{2})^2 \]
This is because:
\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{3}\sqrt{2} = 5 + 2\sqrt{6} \]
Thus,
\[ \sqrt{p} = \sqrt{3} + \sqrt{2} \]

3. **Substitute \(\sqrt{p}\) into the expression**:
We need to evaluate:
\[ \frac{\sqrt{p} - 1}{\sqrt{p}} = \frac{(\sqrt{3} + \sqrt{2}) - 1}{\sqrt{3} + \sqrt{2}} \]

4. **Simplify the expression**:
This can be simplified to:
\[ \frac{\sqrt{3} + \sqrt{2} - 1}{\sqrt{3} + \sqrt{2}} = 1 - \frac{1}{\sqrt{3} + \sqrt{2}} \]

5. **Rationalize \(\frac{1}{\sqrt{3} + \sqrt{2}}\)**:
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3} - \sqrt{2}\):
\[ \frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2} \]

6. **Substitute back into the expression**:
Now substitute back:
\[ 1 - \left(\sqrt{3} - \sqrt{2}\right) = 1 - \sqrt{3} + \sqrt{2} \]

7. **Final Result**:
Therefore, the final value of \(\frac{\sqrt{p} - 1}{\sqrt{p}}\) is:
\[ \sqrt{2} - \sqrt{3} + 1 \]