`(sqrt72 - sqrt18) / sqrt12` is equal to :

Options

A. `sqrt6`
B. `sqrt3//2`
C. `sqrt2 //3`
D. `sqrt6//2`

Answer

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

### Step 1: Simplify \(\sqrt{72}\)
First, we factor \(72\):
\[ 72 = 2^3 \times 3^2 \]
Now, we can simplify \(\sqrt{72}\):
\[ \sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{(2^2 \times 3^2) \times 2} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \]

**Hint:** Factor the number inside the square root to find perfect squares.

### Step 2: Simplify \(\sqrt{18}\)
Next, we factor \(18\):
\[ 18 = 2 \times 3^2 \]
Now, we simplify \(\sqrt{18}\):
\[ \sqrt{18} = \sqrt{2 \times 3^2} = \sqrt{(3^2) \times 2} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \]

**Hint:** Look for pairs of factors to simplify the square root.

### Step 3: Simplify \(\sqrt{12}\)
Now, we factor \(12\):
\[ 12 = 2^2 \times 3 \]
We can simplify \(\sqrt{12}\):
\[ \sqrt{12} = \sqrt{2^2 \times 3} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

**Hint:** Again, find perfect squares to simplify the square root.

### Step 4: Substitute back into the expression
Now we substitute back into the original expression:
\[ \frac{\sqrt{72} - \sqrt{18}}{\sqrt{12}} = \frac{6\sqrt{2} - 3\sqrt{2}}{2\sqrt{3}} \]

### Step 5: Combine like terms in the numerator
Now, we simplify the numerator:
\[ 6\sqrt{2} - 3\sqrt{2} = (6 - 3)\sqrt{2} = 3\sqrt{2} \]

### Step 6: Write the expression as a fraction
Now, we have:
\[ \frac{3\sqrt{2}}{2\sqrt{3}} \]

### Step 7: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by \(\sqrt{3}\):
\[ \frac{3\sqrt{2} \cdot \sqrt{3}}{2\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{6}}{2 \cdot 3} = \frac{3\sqrt{6}}{6} = \frac{\sqrt{6}}{2} \]

### Final Answer
Thus, the final answer is:
\[ \frac{\sqrt{6}}{2} \]

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