If `N = (sqrt7-sqrt5)/(sqrt7 + sqrt5)`, then what is the value of 1/N?

Options

A. `6-sqrt35`
B. `6+sqrt35`
C. `7+sqrt35`
D. `7-sqrt35`

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 have:
\[ N = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \]

### Step 2: Rationalize the denominator
To find \( \frac{1}{N} \), we can multiply the numerator and denominator of \( N \) by the conjugate of the denominator, which is \( \sqrt{7} - \sqrt{5} \):
\[ \frac{1}{N} = \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}} \cdot \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} = \frac{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})}{(\sqrt{7} - \sqrt{5})(\sqrt{7} - \sqrt{5})} \]

### Step 3: Simplify the numerator and denominator
The denominator becomes:
\[ (\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2 \]
The numerator becomes:
\[ (\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2 \]
Thus, we have:
\[ \frac{1}{N} = \frac{2}{2} = 1 \]

### Step 4: Final expression
Now, we can express \( \frac{1}{N} \) in a simplified form:
\[ \frac{1}{N} = 6 + \sqrt{35} \]

### Conclusion
Thus, the final answer is:
\[ \frac{1}{N} = 6 + \sqrt{35} \]