If `N=sqrt""9+sqrt""7`, then what is the value of `(1//N)`

Options

A. `sqrt""9-sqrt""7`
B. `2(sqrt""9-sqrt""7)`
C. `(sqrt""9-sqrt""7)//2`
D. `4(sqrt""9-sqrt""7)`

Answer

To solve the problem, we need to find the value of \( \frac{1}{N} \) where \( N = \sqrt{9} + \sqrt{7} \).

### Step-by-Step Solution:

1. **Identify the value of N**:
\[ N = \sqrt{9} + \sqrt{7} \]
We know that \( \sqrt{9} = 3 \), so:
\[ N = 3 + \sqrt{7} \]

2. **Write the expression for \( \frac{1}{N} \)**:
\[ \frac{1}{N} = \frac{1}{3 + \sqrt{7}} \]

3. **Rationalize the denominator**:
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \( 3 - \sqrt{7} \):
\[ \frac{1}{3 + \sqrt{7}} \cdot \frac{3 - \sqrt{7}}{3 - \sqrt{7}} = \frac{3 - \sqrt{7}}{(3 + \sqrt{7})(3 - \sqrt{7})} \]

4. **Calculate the denominator using the difference of squares**:
The denominator can be simplified using the difference of squares formula:
\[ (3 + \sqrt{7})(3 - \sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9 - 7 = 2 \]

5. **Combine the results**:
Now we can write the expression for \( \frac{1}{N} \):
\[ \frac{1}{N} = \frac{3 - \sqrt{7}}{2} \]

### Final Result:
Thus, the value of \( \frac{1}{N} \) is:
\[ \frac{3 - \sqrt{7}}{2} \]