Given that `sqrt(3) = 1.732, `the value of ` (3 + sqrt(6 ))/( 5 sqrt(3) - 2 sqrt(12) - sqrt(32) + sqrt(50))`is

Options

A. `4.899`
B. `2.551`
C. `1.414`
D. `1.732`

Answer

To solve the expression \((3 + \sqrt{6})/(5\sqrt{3} - 2\sqrt{12} - \sqrt{32} + \sqrt{50})\), we will follow these steps:

### Step 1: Simplify the Denominator
First, we simplify the terms in the denominator:
- \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\)
- \(\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}\)
- \(\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}\)

Now substitute these values back into the denominator:
\[ 5\sqrt{3} - 2\sqrt{12} - \sqrt{32} + \sqrt{50} = 5\sqrt{3} - 2(2\sqrt{3}) - 4\sqrt{2} + 5\sqrt{2} \]
This simplifies to:
\[ 5\sqrt{3} - 4\sqrt{3} + (5\sqrt{2} - 4\sqrt{2}) = (5\sqrt{3} - 4\sqrt{3}) + (5\sqrt{2} - 4\sqrt{2}) = \sqrt{3} + \sqrt{2} \]

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

### Step 3: Rationalize the Denominator
To simplify further, we will rationalize the denominator by multiplying the numerator and denominator by \((\sqrt{3} - \sqrt{2})\):
\[ \frac{(3 + \sqrt{6})(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} \]

### Step 4: Calculate the Denominator
The denominator simplifies to:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]

### Step 5: Calculate the Numerator
Now, we calculate 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:
\[ 3\sqrt{3} - 3\sqrt{2} + \sqrt{18} - \sqrt{12} \]
Where \(\sqrt{18} = 3\sqrt{2}\) and \(\sqrt{12} = 2\sqrt{3}\). Therefore, we have:
\[ 3\sqrt{3} - 3\sqrt{2} + 3\sqrt{2} - 2\sqrt{3} = (3\sqrt{3} - 2\sqrt{3}) + (-3\sqrt{2} + 3\sqrt{2}) = \sqrt{3} \]

### Final Result
Thus, the entire expression simplifies to:
\[ \frac{\sqrt{3}}{1} = \sqrt{3} \]

Given that \(\sqrt{3} \approx 1.732\), the final answer is:
\[ \sqrt{3} \approx 1.732 \]