If `N=((sqrt(7)-sqrt(5)))/((sqrt(7)+sqrt(5)))`, then what is the value of `(1)/(N)` ?

Options

A. `6-sqrt(35)`
B. `6+sqrt(35)`
C. `7+sqrt(35)`
D. `7-sqrt(35)`

Answer

To find the value of \( \frac{1}{N} \) where \( N = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \), we will follow these steps:

### Step 1: Write down the expression for \( N \)
We start with:
\[ N = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \]

### Step 2: Rationalize the denominator
To simplify \( N \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{7} - \sqrt{5} \):
\[ N = \frac{(\sqrt{7} - \sqrt{5})(\sqrt{7} - \sqrt{5})}{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})} \]

### Step 3: Simplify the numerator
The numerator becomes:
\[ (\sqrt{7} - \sqrt{5})^2 = 7 - 2\sqrt{7}\sqrt{5} + 5 = 12 - 2\sqrt{35} \]

### Step 4: Simplify the denominator
The denominator becomes:
\[ (\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2 \]

### Step 5: Combine the results
Now we can write \( N \) as:
\[ N = \frac{12 - 2\sqrt{35}}{2} = 6 - \sqrt{35} \]

### Step 6: Find \( \frac{1}{N} \)
To find \( \frac{1}{N} \), we take the reciprocal:
\[ \frac{1}{N} = \frac{1}{6 - \sqrt{35}} \]

### Step 7: Rationalize \( \frac{1}{N} \)
We again multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{N} = \frac{6 + \sqrt{35}}{(6 - \sqrt{35})(6 + \sqrt{35})} \]

### Step 8: Simplify the denominator
The denominator becomes:
\[ (6)^2 - (\sqrt{35})^2 = 36 - 35 = 1 \]

### Step 9: Final result
Thus, we have:
\[ \frac{1}{N} = 6 + \sqrt{35} \]

### Final Answer
The value of \( \frac{1}{N} \) is:
\[ \boxed{6 + \sqrt{35}} \]