If `(sqrt11-sqrt7)/(sqrt11+sqrt7)=a-bsqrt77,` then (a+b) is equal to:

Options

A. 5
B. 7
C. 11
D. 18

Answer

To solve the equation \((\sqrt{11} - \sqrt{7}) / (\sqrt{11} + \sqrt{7}) = a - b\sqrt{77}\), we will follow these steps:

### Step 1: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \((\sqrt{11} - \sqrt{7})\).

\[ \frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}} \cdot \frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} - \sqrt{7}} = \frac{(\sqrt{11} - \sqrt{7})^2}{(\sqrt{11})^2 - (\sqrt{7})^2} \]

### Step 2: Simplify the denominator
Calculating the denominator:

\[ (\sqrt{11})^2 - (\sqrt{7})^2 = 11 - 7 = 4 \]

### Step 3: Expand the numerator
Now, we expand the numerator:

\[ (\sqrt{11} - \sqrt{7})^2 = 11 - 2\sqrt{11}\sqrt{7} + 7 = 18 - 2\sqrt{77} \]

### Step 4: Combine the results
Now we can substitute back into our equation:

\[ \frac{18 - 2\sqrt{77}}{4} = \frac{18}{4} - \frac{2\sqrt{77}}{4} = \frac{9}{2} - \frac{1}{2}\sqrt{77} \]

### Step 5: Identify \(a\) and \(b\)
From the equation \(\frac{9}{2} - \frac{1}{2}\sqrt{77} = a - b\sqrt{77}\), we can identify:

\[ a = \frac{9}{2}, \quad b = \frac{1}{2} \]

### Step 6: Calculate \(a + b\)
Now, we calculate \(a + b\):

\[ a + b = \frac{9}{2} + \frac{1}{2} = \frac{10}{2} = 5 \]

Thus, the value of \(a + b\) is:

\[ \boxed{5} \]