The value of `sqrt(((sqrt12-sqrt8)(sqrt3+sqrt2))/(5+sqrt(24)))` is-

Options

A. `sqrt6-sqrt2`
B. `sqrt6+sqrt2`
C. `sqrt6-2`
D. `2-sqrt6`

Answer

To solve the expression \( \sqrt{\frac{(\sqrt{12} - \sqrt{8})(\sqrt{3} + \sqrt{2})}{5 + \sqrt{24}}} \), we will follow these steps:

### Step 1: Simplify \( \sqrt{12} \) and \( \sqrt{8} \)
First, we simplify \( \sqrt{12} \) and \( \sqrt{8} \):
\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]
\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
Thus, we can rewrite the expression:
\[ \sqrt{12} - \sqrt{8} = 2\sqrt{3} - 2\sqrt{2} = 2(\sqrt{3} - \sqrt{2}) \]

### Step 2: Substitute back into the expression
Now substitute this back into the original expression:
\[ \sqrt{\frac{(2(\sqrt{3} - \sqrt{2}))(\sqrt{3} + \sqrt{2})}{5 + \sqrt{24}}} \]

### Step 3: Simplify \( \sqrt{24} \)
Next, we simplify \( \sqrt{24} \):
\[ \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} \]
So, we can rewrite the denominator:
\[ 5 + \sqrt{24} = 5 + 2\sqrt{6} \]

### Step 4: Substitute back into the expression
Now, substitute this back into the expression:
\[ \sqrt{\frac{2(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}{5 + 2\sqrt{6}}} \]

### Step 5: Simplify the numerator
The numerator can be simplified using the difference of squares:
\[ (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = 3 - 2 = 1 \]
Thus, the numerator becomes:
\[ 2 \cdot 1 = 2 \]

### Step 6: Rewrite the expression
Now we can rewrite the expression:
\[ \sqrt{\frac{2}{5 + 2\sqrt{6}}} \]

### Step 7: Rationalize the denominator
To simplify further, we can rationalize the denominator:
Multiply the numerator and denominator by the conjugate of the denominator:
\[ \sqrt{\frac{2(5 - 2\sqrt{6})}{(5 + 2\sqrt{6})(5 - 2\sqrt{6})}} \]
Calculating the denominator:
\[ (5 + 2\sqrt{6})(5 - 2\sqrt{6}) = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]
Thus, the expression simplifies to:
\[ \sqrt{2(5 - 2\sqrt{6})} = \sqrt{10 - 4\sqrt{6}} \]

### Step 8: Final result
The final result is:
\[ \sqrt{10 - 4\sqrt{6}} \]