If `N=((sqrt(8)+sqrt(6)))/((sqrt(8)-sqrt(6)))`, then what is the value of `N-((1)/(N))` ?

Options

A. `8sqrt(3)`
B. `14`
C. `7`
D. `0`

Answer

To solve the problem, we start with the expression for \( N \):

\[ N = \frac{\sqrt{8} + \sqrt{6}}{\sqrt{8} - \sqrt{6}} \]

### Step 1: Rationalize the denominator

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

\[ N = \frac{(\sqrt{8} + \sqrt{6})(\sqrt{8} + \sqrt{6})}{(\sqrt{8} - \sqrt{6})(\sqrt{8} + \sqrt{6})} \]

### Step 2: Expand the numerator and denominator

Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) for the numerator:

\[ \text{Numerator: } (\sqrt{8} + \sqrt{6})^2 = 8 + 6 + 2\sqrt{8 \cdot 6} = 14 + 2\sqrt{48} \]

For the denominator, we use \( (a - b)(a + b) = a^2 - b^2 \):

\[ \text{Denominator: } (\sqrt{8})^2 - (\sqrt{6})^2 = 8 - 6 = 2 \]

### Step 3: Combine the results

Now substituting back into the expression for \( N \):

\[ N = \frac{14 + 2\sqrt{48}}{2} \]

### Step 4: Simplify \( N \)

We can simplify this further:

\[ N = \frac{14}{2} + \frac{2\sqrt{48}}{2} = 7 + \sqrt{48} \]

Since \( \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \):

\[ N = 7 + 4\sqrt{3} \]

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

To find \( \frac{1}{N} \), we can use the formula for the reciprocal of a sum:

\[ \frac{1}{N} = \frac{1}{7 + 4\sqrt{3}} \]

Multiplying the numerator and denominator by the conjugate:

\[ \frac{1}{N} = \frac{7 - 4\sqrt{3}}{(7 + 4\sqrt{3})(7 - 4\sqrt{3})} \]

Calculating the denominator:

\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 49 - 48 = 1 \]

Thus,

\[ \frac{1}{N} = 7 - 4\sqrt{3} \]

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

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

\[ N - \frac{1}{N} = (7 + 4\sqrt{3}) - (7 - 4\sqrt{3}) \]

This simplifies to:

\[ N - \frac{1}{N} = 7 + 4\sqrt{3} - 7 + 4\sqrt{3} = 8\sqrt{3} \]

### Final Answer

Thus, the value of \( N - \frac{1}{N} \) is:

\[ \boxed{8\sqrt{3}} \]