Simplify: `(4sqrt(18))/(sqrt(12))-(8sqrt(75))/(sqrt(32))+(9sqrt(2))/(sqrt(3))`

Options

A. `0`
B. `-1`
C. `1`
D. `2`

Answer

To simplify the expression \((4\sqrt{18})/(\sqrt{12}) - (8\sqrt{75})/(\sqrt{32}) + (9\sqrt{2})/(\sqrt{3})\), we will follow these steps:

### Step 1: Simplify each square root term
1. **Simplify \(\sqrt{18}\)**:
\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \]
Thus, \(4\sqrt{18} = 4 \times 3\sqrt{2} = 12\sqrt{2}\).

2. **Simplify \(\sqrt{12}\)**:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

3. **Simplify \(\sqrt{75}\)**:
\[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \]
Thus, \(8\sqrt{75} = 8 \times 5\sqrt{3} = 40\sqrt{3}\).

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

5. **Simplify \(\sqrt{2}\) and \(\sqrt{3}\)**:
\[ \sqrt{2} \text{ and } \sqrt{3} \text{ remain as they are.} \]

### Step 2: Rewrite the expression
Now substituting the simplified terms back into the expression:
\[ \frac{12\sqrt{2}}{2\sqrt{3}} - \frac{40\sqrt{3}}{4\sqrt{2}} + \frac{9\sqrt{2}}{\sqrt{3}} \]

### Step 3: Simplify each fraction
1. **First term**:
\[ \frac{12\sqrt{2}}{2\sqrt{3}} = \frac{12}{2} \cdot \frac{\sqrt{2}}{\sqrt{3}} = 6\frac{\sqrt{2}}{\sqrt{3}} = 6\frac{\sqrt{6}}{3} = 2\sqrt{6} \]

2. **Second term**:
\[ \frac{40\sqrt{3}}{4\sqrt{2}} = \frac{40}{4} \cdot \frac{\sqrt{3}}{\sqrt{2}} = 10\frac{\sqrt{3}}{\sqrt{2}} = 10\frac{\sqrt{6}}{2} = 5\sqrt{6} \]

3. **Third term**:
\[ \frac{9\sqrt{2}}{\sqrt{3}} = 9\frac{\sqrt{2}}{\sqrt{3}} = 9\frac{\sqrt{6}}{3} = 3\sqrt{6} \]

### Step 4: Combine the terms
Now we can combine the simplified terms:
\[ 2\sqrt{6} - 5\sqrt{6} + 3\sqrt{6} = (2 - 5 + 3)\sqrt{6} = 0\sqrt{6} = 0 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{0} \]