`(sqrt(72) - sqrt(18)) div sqrt(12)` is equal to :

Options

A. `sqrt(6)`
B. `sqrt(3)//2`
C. `sqrt(2)//3`
D. `sqrt(6)//2`

Answer

To solve the expression \((\sqrt{72} - \sqrt{18}) \div \sqrt{12}\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Expression**:
\[ \frac{\sqrt{72} - \sqrt{18}}{\sqrt{12}} \]

2. **Separate the Fractions**:
We can separate the terms in the numerator:
\[ = \frac{\sqrt{72}}{\sqrt{12}} - \frac{\sqrt{18}}{\sqrt{12}} \]

3. **Combine the Square Roots**:
Using the property \(\frac{\sqrt{m}}{\sqrt{n}} = \sqrt{\frac{m}{n}}\):
\[ = \sqrt{\frac{72}{12}} - \sqrt{\frac{18}{12}} \]

4. **Simplify the Fractions Inside the Square Roots**:
Calculate \(\frac{72}{12}\) and \(\frac{18}{12}\):
\[ \frac{72}{12} = 6 \quad \text{and} \quad \frac{18}{12} = \frac{3}{2} \]
Thus, we have:
\[ = \sqrt{6} - \sqrt{\frac{3}{2}} \]

5. **Simplify \(\sqrt{\frac{3}{2}}\)**:
Rewrite \(\sqrt{\frac{3}{2}}\) as \(\frac{\sqrt{3}}{\sqrt{2}}\):
\[ = \sqrt{6} - \frac{\sqrt{3}}{\sqrt{2}} \]

6. **Find a Common Denominator**:
To combine the terms, we can express \(\sqrt{6}\) in terms of \(\sqrt{2}\):
\[ \sqrt{6} = \frac{\sqrt{6} \cdot \sqrt{2}}{\sqrt{2}} = \frac{\sqrt{12}}{\sqrt{2}} \]
Therefore:
\[ = \frac{\sqrt{12}}{\sqrt{2}} - \frac{\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{12} - \sqrt{3}}{\sqrt{2}} \]

7. **Simplify the Numerator**:
We can simplify \(\sqrt{12}\):
\[ \sqrt{12} = 2\sqrt{3} \]
Thus:
\[ = \frac{2\sqrt{3} - \sqrt{3}}{\sqrt{2}} = \frac{(2 - 1)\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{3}}{\sqrt{2}} \]

8. **Rationalize the Denominator**:
Multiply the numerator and denominator by \(\sqrt{2}\):
\[ = \frac{\sqrt{3} \cdot \sqrt{2}}{2} = \frac{\sqrt{6}}{2} \]

### Final Answer:
\[ \frac{\sqrt{6}}{2} \]