The value of `sqrt(6+2sqrt(3)+2sqrt(2)+2sqrt(6))-(1)/(sqrt(5-2sqrt(6)))` is

Options

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

Answer

To find the value of \(\sqrt{6 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6}} - \frac{1}{\sqrt{5 - 2\sqrt{6}}}\), follow these steps:

### Step 1: Simplify the Square Root Expression
Consider the expression inside the square root: \(6 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6}\).

Notice that:
\[ 6 = 1 + 2 + 3 \]
So, rewrite the expression as:
\[ 6 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6} = 1 + 2 + 3 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6} \]

This can be recognized as the expansion of:
\[ (1 + \sqrt{3} + \sqrt{2})^2 \]

### Step 2: Verify the Identity
Expand \((1 + \sqrt{3} + \sqrt{2})^2\):
\[ (1 + \sqrt{3} + \sqrt{2})^2 = 1^2 + (\sqrt{3})^2 + (\sqrt{2})^2 + 2 \cdot 1 \cdot \sqrt{3} + 2 \cdot 1 \cdot \sqrt{2} + 2 \cdot \sqrt{3} \cdot \sqrt{2} \]
\[ = 1 + 3 + 2 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6} \]
\[ = 6 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6} \]

Thus:
\[ \sqrt{6 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6}} = 1 + \sqrt{3} + \sqrt{2} \]

### Step 3: Simplify the Denominator
Consider the denominator: \(\sqrt{5 - 2\sqrt{6}}\).

Recognize that:
\[ 5 - 2\sqrt{6} = (\sqrt{2} - \sqrt{3})^2 \]

### Step 4: Verify the Identity
Expand \((\sqrt{2} - \sqrt{3})^2\):
\[ (\sqrt{2} - \sqrt{3})^2 = (\sqrt{2})^2 + (\sqrt{3})^2 - 2 \cdot \sqrt{2} \cdot \sqrt{3} \]
\[ = 2 + 3 - 2\sqrt{6} \]
\[ = 5 - 2\sqrt{6} \]

Thus:
\[ \sqrt{5 - 2\sqrt{6}} = \sqrt{2} - \sqrt{3} \]

### Step 5: Substitute and Simplify
Now substitute back into the original expression:
\[ 1 + \sqrt{3} + \sqrt{2} - \frac{1}{\sqrt{2} - \sqrt{3}} \]

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

Thus:
\[ 1 + \sqrt{3} + \sqrt{2} - (-(\sqrt{2} + \sqrt{3})) = 1 + \sqrt{3} + \sqrt{2} + \sqrt{2} + \sqrt{3} \]
\[ = 1 + 2\sqrt{3} + 2\sqrt{2} \]

So, the value of the given expression is:
\[ \boxed{1} \]