The expression `sqrt(12+6sqrt(3))+sqrt(12-6sqrt(3))` simplifies to

Options

A. 4
B. `2sqrt(3)`
C. `3sqrt(3)`
D. 6

Answer

To simplify the expression \( \sqrt{12 + 6\sqrt{3}} + \sqrt{12 - 6\sqrt{3}} \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ x = \sqrt{12 + 6\sqrt{3}} + \sqrt{12 - 6\sqrt{3}} \]

### Step 2: Square Both Sides
To eliminate the square roots, we square both sides:
\[ x^2 = \left(\sqrt{12 + 6\sqrt{3}} + \sqrt{12 - 6\sqrt{3}}\right)^2 \]

### Step 3: Expand the Right Side
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), we expand:
\[ x^2 = \left(\sqrt{12 + 6\sqrt{3}}\right)^2 + 2\sqrt{12 + 6\sqrt{3}}\sqrt{12 - 6\sqrt{3}} + \left(\sqrt{12 - 6\sqrt{3}}\right)^2 \]
This simplifies to:
\[ x^2 = (12 + 6\sqrt{3}) + (12 - 6\sqrt{3}) + 2\sqrt{(12 + 6\sqrt{3})(12 - 6\sqrt{3})} \]

### Step 4: Simplify the Expression
Combining the terms gives:
\[ x^2 = 12 + 12 + 2\sqrt{(12 + 6\sqrt{3})(12 - 6\sqrt{3})} \]
\[ x^2 = 24 + 2\sqrt{(12)^2 - (6\sqrt{3})^2} \]

### Step 5: Calculate the Product Inside the Square Root
Calculating \( (12)^2 - (6\sqrt{3})^2 \):
\[ (12)^2 = 144 \quad \text{and} \quad (6\sqrt{3})^2 = 36 \times 3 = 108 \]
Thus,
\[ x^2 = 24 + 2\sqrt{144 - 108} = 24 + 2\sqrt{36} \]
\[ \sqrt{36} = 6 \]
So,
\[ x^2 = 24 + 2 \times 6 = 24 + 12 = 36 \]

### Step 6: Take the Square Root
Taking the square root of both sides:
\[ x = \sqrt{36} = 6 \]

### Conclusion
The simplified expression is:
\[ \sqrt{12 + 6\sqrt{3}} + \sqrt{12 - 6\sqrt{3}} = 6 \]