If `N = ( sqrt7 - sqrt3) //(sqrt7 + sqrt3),` then what is the value of N + (1/N)?

Options

A. `2 sqrt2`
B. 5
C. 10
D. 13

Answer

To solve the problem, we need to find the value of \( N + \frac{1}{N} \) where \( N = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}} \).

### Step-by-Step Solution:

1. **Calculate \( N \)**:
\[ N = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}} \]

2. **Find \( \frac{1}{N} \)**:
To find \( \frac{1}{N} \), we can take the reciprocal of \( N \):
\[ \frac{1}{N} = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \]

3. **Calculate \( N + \frac{1}{N} \)**:
Now we need to add \( N \) and \( \frac{1}{N} \):
\[ N + \frac{1}{N} = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}} + \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \]

4. **Find a common denominator**:
The common denominator for the two fractions is \( (\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) \):
\[ N + \frac{1}{N} = \frac{(\sqrt{7} - \sqrt{3})^2 + (\sqrt{7} + \sqrt{3})^2}{(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3})} \]

5. **Expand the numerators**:
- For \( (\sqrt{7} - \sqrt{3})^2 \):
\[ (\sqrt{7})^2 - 2\sqrt{7}\sqrt{3} + (\sqrt{3})^2 = 7 - 2\sqrt{21} + 3 = 10 - 2\sqrt{21} \]
- For \( (\sqrt{7} + \sqrt{3})^2 \):
\[ (\sqrt{7})^2 + 2\sqrt{7}\sqrt{3} + (\sqrt{3})^2 = 7 + 2\sqrt{21} + 3 = 10 + 2\sqrt{21} \]

6. **Combine the results**:
\[ N + \frac{1}{N} = \frac{(10 - 2\sqrt{21}) + (10 + 2\sqrt{21})}{(\sqrt{7})^2 - (\sqrt{3})^2} \]
\[ = \frac{20}{7 - 3} = \frac{20}{4} = 5 \]

### Final Result:
Thus, the value of \( N + \frac{1}{N} \) is \( 5 \).