Square root of `6 + sqrt(12) - sqrt(24) - sqrt(8)` is

Options

A. `sqrt(3) + sqrt(2) - 1`
B. `sqrt(3) + 1 - sqrt(2)`
C. `1 + sqrt(2) - sqrt(3)`
D. None of these

Answer

To solve the expression \( \sqrt{6 + \sqrt{12} - \sqrt{24} - \sqrt{8}} \), we will simplify the terms inside the square root step by step.

### Step 1: Simplify the square roots
First, we will simplify the square roots in the expression:

- \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \)
- \( \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} \)
- \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \)

Now, we can rewrite the expression:

\[ \sqrt{6 + 2\sqrt{3} - 2\sqrt{6} - 2\sqrt{2}} \]

### Step 2: Rearranging the expression
Next, we rearrange the terms inside the square root:

\[ \sqrt{6 - 2\sqrt{6} + 2\sqrt{3} - 2\sqrt{2}} \]

### Step 3: Grouping the terms
We can group the terms to see if we can factor them:

\[ \sqrt{(6 - 2\sqrt{6}) + (2\sqrt{3} - 2\sqrt{2})} \]

### Step 4: Factor the first group
The first group \( 6 - 2\sqrt{6} \) can be factored:

\[ 6 - 2\sqrt{6} = (\sqrt{6} - 1)^2 \]

### Step 5: Factor the second group
The second group \( 2\sqrt{3} - 2\sqrt{2} \) can be factored as:

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

### Step 6: Combine the results
Now we can combine the results:

\[ \sqrt{(\sqrt{6} - 1)^2 + 2(\sqrt{3} - \sqrt{2})} \]

### Step 7: Final simplification
Now we can express the entire expression under the square root:

\[ \sqrt{(\sqrt{6} - 1)^2 + 2(\sqrt{3} - \sqrt{2})} \]

### Conclusion
The final expression simplifies to:

\[ \sqrt{3} + 1 - \sqrt{2} \]

Thus, the answer is:

\[ \sqrt{3} + 1 - \sqrt{2} \]