`1+(4sqrt(3))/(2-sqrt(2)) -30/(4sqrt(3)-sqrt(18)) - sqrt(18)/(3+2sqrt(3))` is simplified to:

Options

A. 0
B. 1
C. `sqrt(2)`
D. `sqrt(3)`

Answer

To simplify the expression \( 1 + \frac{4\sqrt{3}}{2 - \sqrt{2}} - \frac{30}{4\sqrt{3} - \sqrt{18}} - \frac{\sqrt{18}}{3 + 2\sqrt{3}} \), we will follow these steps:

### Step 1: Simplify each term separately

1. **Simplifying \( \frac{4\sqrt{3}}{2 - \sqrt{2}} \)**:
- Multiply the numerator and denominator by the conjugate of the denominator, \( 2 + \sqrt{2} \):
\[ \frac{4\sqrt{3}(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} \]
- The denominator simplifies to:
\[ 2^2 - (\sqrt{2})^2 = 4 - 2 = 2 \]
- Thus, we have:
\[ \frac{4\sqrt{3}(2 + \sqrt{2})}{2} = 2\sqrt{3}(2 + \sqrt{2}) = 4\sqrt{3} + 2\sqrt{6} \]

2. **Simplifying \( -\frac{30}{4\sqrt{3} - \sqrt{18}} \)**:
- Note that \( \sqrt{18} = 3\sqrt{2} \), so we rewrite it as:
\[ -\frac{30}{4\sqrt{3} - 3\sqrt{2}} \]
- Again, multiply by the conjugate \( 4\sqrt{3} + 3\sqrt{2} \):
\[ -\frac{30(4\sqrt{3} + 3\sqrt{2})}{(4\sqrt{3} - 3\sqrt{2})(4\sqrt{3} + 3\sqrt{2})} \]
- The denominator simplifies to:
\[ (4\sqrt{3})^2 - (3\sqrt{2})^2 = 48 - 18 = 30 \]
- Thus, we have:
\[ -\frac{30(4\sqrt{3} + 3\sqrt{2})}{30} = -(4\sqrt{3} + 3\sqrt{2}) = -4\sqrt{3} - 3\sqrt{2} \]

3. **Simplifying \( -\frac{\sqrt{18}}{3 + 2\sqrt{3}} \)**:
- Rewrite \( \sqrt{18} = 3\sqrt{2} \):
\[ -\frac{3\sqrt{2}}{3 + 2\sqrt{3}} \]
- Multiply by the conjugate \( 3 - 2\sqrt{3} \):
\[ -\frac{3\sqrt{2}(3 - 2\sqrt{3})}{(3 + 2\sqrt{3})(3 - 2\sqrt{3})} \]
- The denominator simplifies to:
\[ 3^2 - (2\sqrt{3})^2 = 9 - 12 = -3 \]
- Thus, we have:
\[ -\frac{3\sqrt{2}(3 - 2\sqrt{3})}{-3} = \sqrt{2}(3 - 2\sqrt{3}) = 3\sqrt{2} - 2\sqrt{6} \]

### Step 2: Combine all the simplified terms

Now we combine all the simplified terms:
\[ 1 + (4\sqrt{3} + 2\sqrt{6}) - (4\sqrt{3} + 3\sqrt{2}) + (3\sqrt{2} - 2\sqrt{6}) \]

### Step 3: Simplify the expression

Combine like terms:
- The \( 4\sqrt{3} \) terms cancel out:
\[ 1 + 2\sqrt{6} - 3\sqrt{2} + 3\sqrt{2} - 2\sqrt{6} = 1 + 0 + 0 = 1 \]

### Final Answer
The expression simplifies to:
\[ \boxed{1} \]