Value of `(3+sqrt6)/(5sqrt3-2sqrt12-sqrt32+sqrt50)` is

Options

A. `1/sqrt3`
B. `sqrt3`
C. `1/3`
D. 3

Answer

To solve the expression \(\frac{3+\sqrt{6}}{5\sqrt{3}-2\sqrt{12}-\sqrt{32}+\sqrt{50}}\), we will simplify the denominator first and then rationalize the fraction.

### Step 1: Simplify the Denominator
The denominator is \(5\sqrt{3} - 2\sqrt{12} - \sqrt{32} + \sqrt{50}\).

1. **Simplify \(\sqrt{12}\)**:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]
Therefore, \(-2\sqrt{12} = -2 \cdot 2\sqrt{3} = -4\sqrt{3}\).

2. **Simplify \(\sqrt{32}\)**:
\[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \]
Therefore, \(-\sqrt{32} = -4\sqrt{2}\).

3. **Simplify \(\sqrt{50}\)**:
\[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]

Now, substituting these back into the denominator:
\[ 5\sqrt{3} - 4\sqrt{3} - 4\sqrt{2} + 5\sqrt{2} \]
Combine like terms:
\[ (5\sqrt{3} - 4\sqrt{3}) + (-4\sqrt{2} + 5\sqrt{2}) = \sqrt{3} + \sqrt{2} \]

### Step 2: Rewrite the Expression
Now, we can rewrite the expression:
\[ \frac{3+\sqrt{6}}{\sqrt{3}+\sqrt{2}} \]

### Step 3: Rationalize the Denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{3}-\sqrt{2}\):
\[ \frac{(3+\sqrt{6})(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})} \]

### Step 4: Simplify the Denominator
Using the difference of squares:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]

### Step 5: Expand the Numerator
Now, we expand the numerator:
\[ (3+\sqrt{6})(\sqrt{3}-\sqrt{2}) = 3\sqrt{3} - 3\sqrt{2} + \sqrt{6}\sqrt{3} - \sqrt{6}\sqrt{2} \]
This simplifies to:
\[ 3\sqrt{3} - 3\sqrt{2} + \sqrt{18} - \sqrt{12} \]
Now, simplify \(\sqrt{18}\) and \(\sqrt{12}\):
\[ \sqrt{18} = 3\sqrt{2}, \quad \sqrt{12} = 2\sqrt{3} \]
Substituting these values back:
\[ 3\sqrt{3} - 3\sqrt{2} + 3\sqrt{2} - 2\sqrt{3} \]
Combine like terms:
\[ (3\sqrt{3} - 2\sqrt{3}) + (-3\sqrt{2} + 3\sqrt{2}) = \sqrt{3} + 0 = \sqrt{3} \]

### Final Answer
Thus, the value of the expression is:
\[ \sqrt{3} \]