The value of `sqrt(5+2sqrt(6))-(1)/(sqrt(5+2sqrt(6)))` is :

Options

A. `2sqrt(2)`
B. `2sqrt(3)`
C. `1+sqrt(5)`
D. `sqrt(5)-1`

Answer

To solve the expression \( \sqrt{5 + 2\sqrt{6}} - \frac{1}{\sqrt{5 + 2\sqrt{6}}} \), we can follow these steps:

### Step 1: Simplify \( \sqrt{5 + 2\sqrt{6}} \)

We start by rewriting \( \sqrt{5 + 2\sqrt{6}} \). We can express \( 5 + 2\sqrt{6} \) in a form that allows us to take the square root more easily. We notice that:

\[ 5 + 2\sqrt{6} = (\sqrt{3} + \sqrt{2})^2 \]

This is because:

\[ (\sqrt{3} + \sqrt{2})^2 = (\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{2}) + (\sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \]

Thus, we have:

\[ \sqrt{5 + 2\sqrt{6}} = \sqrt{(\sqrt{3} + \sqrt{2})^2} = \sqrt{3} + \sqrt{2} \]

### Step 2: Substitute back into the expression

Now we substitute this back into our original expression:

\[ \sqrt{5 + 2\sqrt{6}} - \frac{1}{\sqrt{5 + 2\sqrt{6}}} = (\sqrt{3} + \sqrt{2}) - \frac{1}{\sqrt{3} + \sqrt{2}} \]

### Step 3: Simplify \( \frac{1}{\sqrt{3} + \sqrt{2}} \)

To simplify \( \frac{1}{\sqrt{3} + \sqrt{2}} \), we can multiply the numerator and the denominator by the conjugate \( \sqrt{3} - \sqrt{2} \):

\[ \frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2} \]

### Step 4: Combine the terms

Now we can substitute this back into our expression:

\[ \sqrt{3} + \sqrt{2} - (\sqrt{3} - \sqrt{2}) = \sqrt{3} + \sqrt{2} - \sqrt{3} + \sqrt{2} \]

This simplifies to:

\[ 2\sqrt{2} \]

### Final Answer

Thus, the value of the expression \( \sqrt{5 + 2\sqrt{6}} - \frac{1}{\sqrt{5 + 2\sqrt{6}}} \) is:

\[ \boxed{2\sqrt{2}} \]