If `P=((sqrt(7)-sqrt(6)))/((sqrt(7)+sqrt(6)))`, then what is the value of `(P+(1)/(P))` ?

Options

A. 12
B. 13
C. 24
D. 26

Answer

To solve the problem, we need to find the value of \( P + \frac{1}{P} \) where \( P = \frac{\sqrt{7} - \sqrt{6}}{\sqrt{7} + \sqrt{6}} \).

### Step 1: Rationalize the denominator of \( P \)

To simplify \( P \), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{7} - \sqrt{6} \):

\[ P = \frac{(\sqrt{7} - \sqrt{6})(\sqrt{7} - \sqrt{6})}{(\sqrt{7} + \sqrt{6})(\sqrt{7} - \sqrt{6})} \]

### Step 2: Simplify the denominator using the difference of squares

The denominator simplifies to:

\[ (\sqrt{7})^2 - (\sqrt{6})^2 = 7 - 6 = 1 \]

### Step 3: Simplify the numerator

The numerator becomes:

\[ (\sqrt{7} - \sqrt{6})^2 = (\sqrt{7})^2 - 2\sqrt{7}\sqrt{6} + (\sqrt{6})^2 = 7 - 2\sqrt{42} + 6 = 13 - 2\sqrt{42} \]

### Step 4: Substitute back to find \( P \)

Thus, we have:

\[ P = \frac{13 - 2\sqrt{42}}{1} = 13 - 2\sqrt{42} \]

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

Next, we need to find \( \frac{1}{P} \):

\[ \frac{1}{P} = \frac{1}{13 - 2\sqrt{42}} \]

To rationalize this, we multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1}{P} = \frac{13 + 2\sqrt{42}}{(13 - 2\sqrt{42})(13 + 2\sqrt{42})} \]

### Step 6: Simplify the denominator of \( \frac{1}{P} \)

The denominator simplifies to:

\[ (13)^2 - (2\sqrt{42})^2 = 169 - 168 = 1 \]

Thus,

\[ \frac{1}{P} = 13 + 2\sqrt{42} \]

### Step 7: Calculate \( P + \frac{1}{P} \)

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

\[ P + \frac{1}{P} = (13 - 2\sqrt{42}) + (13 + 2\sqrt{42}) = 13 + 13 = 26 \]

### Final Answer

Thus, the value of \( P + \frac{1}{P} \) is:

\[ \boxed{26} \]