What is the value of `(sqrt(6)+sqrt(5))/(sqrt(6)-sqrt(5))`?

Options

A. `11– 2sqrt(30) `
B. `11 + 2 sqrt(30)`
C. `6 + 2 sqrt(30)`
D. `6-2 sqrt(30)`

Answer

To solve the expression \((\sqrt{6} + \sqrt{5}) / (\sqrt{6} - \sqrt{5})\), we can use the method of rationalization. Here’s the step-by-step solution:

### Step 1: Write the expression
We start with the expression:
\[ \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} - \sqrt{5}} \]

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{6} + \sqrt{5})\):
\[ \frac{(\sqrt{6} + \sqrt{5})(\sqrt{6} + \sqrt{5})}{(\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5})} \]

### 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: Simplify the denominator
The denominator simplifies using the difference of squares:
\[ (\sqrt{6})^2 - (\sqrt{5})^2 = 6 - 5 = 1 \]

### Step 5: Combine the results
Now we can combine the results from the numerator and denominator:
\[ \frac{11 + 2\sqrt{30}}{1} = 11 + 2\sqrt{30} \]

### Final Answer
Thus, the value of \(\frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} - \sqrt{5}}\) is:
\[ 11 + 2\sqrt{30} \]

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