If `1/(sqrt(2011+sqrt(2011^2-1)))=sqrtm-sqrtn` where m and n are positive integers , what is the value of m + n.

Answer

To solve the equation \( \frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} = \sqrt{m} - \sqrt{n} \), where \( m \) and \( n \) are positive integers, we will follow these steps:

### Step 1: Simplify the Left Side
We start with the expression on the left side:
\[ \frac{1}{\sqrt{2011 + \sqrt{2011^2 - 1}}} \]

### Step 2: Rationalize the Denominator
To simplify this expression, we will rationalize the denominator. We multiply the numerator and denominator by \( \sqrt{2011 - \sqrt{2011^2 - 1}} \):
\[ \frac{\sqrt{2011 - \sqrt{2011^2 - 1}}}{\sqrt{(2011 + \sqrt{2011^2 - 1})(2011 - \sqrt{2011^2 - 1})}} \]

### Step 3: Simplify the Denominator
Using the difference of squares:
\[ (2011 + \sqrt{2011^2 - 1})(2011 - \sqrt{2011^2 - 1}) = 2011^2 - (2011^2 - 1) = 1 \]
Thus, the expression simplifies to:
\[ \sqrt{2011 - \sqrt{2011^2 - 1}} \]

### Step 4: Set the Expression Equal to the Right Side
Now we have:
\[ \sqrt{2011 - \sqrt{2011^2 - 1}} = \sqrt{m} - \sqrt{n} \]

### Step 5: Square Both Sides
Squaring both sides gives:
\[ 2011 - \sqrt{2011^2 - 1} = m + n - 2\sqrt{mn} \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ \sqrt{2011^2 - 1} = m + n - 2011 + 2\sqrt{mn} \]

### Step 7: Identify \( m \) and \( n \)
To express \( \sqrt{2011 - \sqrt{2011^2 - 1}} \) in the form \( \sqrt{m} - \sqrt{n} \), we can rewrite:
\[ 2011 - \sqrt{2011^2 - 1} = 2011 - \sqrt{(2011 - 1)(2011 + 1)} = 2011 - \sqrt{2010 \cdot 2012} \]
This can be expressed as:
\[ 2011 - \sqrt{2011^2 - 1} = 1006 + 1005 - 2\sqrt{1006 \cdot 1005} \]
Thus, we have \( m = 1006 \) and \( n = 1005 \).

### Step 8: Calculate \( m + n \)
Finally, we find:
\[ m + n = 1006 + 1005 = 2011 \]

### Final Answer
The value of \( m + n \) is \( \boxed{2011} \).

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