If `(sqrt7-sqrt5)/(sqrt7+sqrt5)=a+bsqrt(35)`, then the value of (a - b) is

Options

A. 5
B. 6
C. 8
D. None of these

Answer

To solve the equation \(\frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} = a + b\sqrt{35}\), we will follow these steps:

### Step 1: Rationalize the Left-Hand Side
We start with the left-hand side:
\[ \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{7} - \sqrt{5}\):
\[ \frac{(\sqrt{7} - \sqrt{5})(\sqrt{7} - \sqrt{5})}{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})} \]

### Step 2: Simplify the Numerator
Now, we simplify the numerator:
\[ (\sqrt{7} - \sqrt{5})^2 = \sqrt{7}^2 - 2\sqrt{7}\sqrt{5} + \sqrt{5}^2 = 7 - 2\sqrt{35} + 5 = 12 - 2\sqrt{35} \]

### Step 3: Simplify the Denominator
Next, we simplify the denominator using the difference of squares:
\[ (\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2 \]

### Step 4: Combine the Results
Now we can combine the results:
\[ \frac{12 - 2\sqrt{35}}{2} = \frac{12}{2} - \frac{2\sqrt{35}}{2} = 6 - \sqrt{35} \]

### Step 5: Set Equal to Right-Hand Side
Now we have:
\[ 6 - \sqrt{35} = a + b\sqrt{35} \]
From this, we can equate the coefficients:
- \(a = 6\)
- \(b = -1\)

### Step 6: Calculate \(a - b\)
Finally, we calculate \(a - b\):
\[ a - b = 6 - (-1) = 6 + 1 = 7 \]

Thus, the value of \(a - b\) is \(\boxed{7}\).