If `N=((sqrt(6)-sqrt(5)))/((sqrt(6)+sqrt(5)))` then what is the value of `N+((1)/(N))` ?

Options

A. a) `10`
B. b) `11`
C. c) `12`
D. d) `22`

Answer

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

### Step 1: Rationalize \( N \)

To simplify \( N \), we can multiply the numerator and denominator by the conjugate of the denominator:

\[ N = \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} + \sqrt{5}} \cdot \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}} \]

### Step 2: Simplify the denominator

The denominator becomes:

\[ (\sqrt{6} + \sqrt{5})(\sqrt{6} - \sqrt{5}) = (\sqrt{6})^2 - (\sqrt{5})^2 = 6 - 5 = 1 \]

### Step 3: Simplify the numerator

The numerator becomes:

\[ (\sqrt{6} - \sqrt{5})^2 = (\sqrt{6})^2 - 2(\sqrt{6})(\sqrt{5}) + (\sqrt{5})^2 = 6 - 2\sqrt{30} + 5 = 11 - 2\sqrt{30} \]

### Step 4: Combine the results

Thus, we have:

\[ N = 11 - 2\sqrt{30} \]

### Step 5: Find \( \frac{1}{N} \)

To find \( \frac{1}{N} \), we can use the fact that:

\[ \frac{1}{N} = \frac{1}{11 - 2\sqrt{30}} \]

Rationalizing \( \frac{1}{N} \):

\[ \frac{1}{N} = \frac{11 + 2\sqrt{30}}{(11 - 2\sqrt{30})(11 + 2\sqrt{30})} \]

Calculating the denominator:

\[ (11 - 2\sqrt{30})(11 + 2\sqrt{30}) = 11^2 - (2\sqrt{30})^2 = 121 - 120 = 1 \]

Thus, we have:

\[ \frac{1}{N} = 11 + 2\sqrt{30} \]

### Step 6: Calculate \( N + \frac{1}{N} \)

Now we can find \( N + \frac{1}{N} \):

\[ N + \frac{1}{N} = (11 - 2\sqrt{30}) + (11 + 2\sqrt{30}) = 11 + 11 = 22 \]

### Final Answer

The value of \( N + \frac{1}{N} \) is:

\[ \boxed{22} \]