Value of `sqrt(28 - 6sqrt(3)) + sqrt(28 + 6sqrt(3))` is

Options

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

Answer

To solve the expression \( \sqrt{28 - 6\sqrt{3}} + \sqrt{28 + 6\sqrt{3}} \), we can simplify each square root separately.

### Step 1: Rewrite the expressions under the square roots
We start with the two square roots:
\[ \sqrt{28 - 6\sqrt{3}} \quad \text{and} \quad \sqrt{28 + 6\sqrt{3}} \]

### Step 2: Express \( 28 \) in a suitable form
Notice that \( 28 \) can be expressed as \( (3\sqrt{3})^2 + 1^2 \):
\[ 28 = (3\sqrt{3})^2 + 1^2 \]

### Step 3: Identify \( a \) and \( b \) for the square root expressions
We can rewrite \( 28 - 6\sqrt{3} \) and \( 28 + 6\sqrt{3} \) using the identity \( a^2 - 2ab + b^2 = (a - b)^2 \) and \( a^2 + 2ab + b^2 = (a + b)^2 \):
- Here, let \( a = 3\sqrt{3} \) and \( b = 1 \).

### Step 4: Simplify \( \sqrt{28 - 6\sqrt{3}} \)
Using the identity:
\[ 28 - 6\sqrt{3} = (3\sqrt{3} - 1)^2 \]
Thus,
\[ \sqrt{28 - 6\sqrt{3}} = 3\sqrt{3} - 1 \]

### Step 5: Simplify \( \sqrt{28 + 6\sqrt{3}} \)
Similarly,
\[ 28 + 6\sqrt{3} = (3\sqrt{3} + 1)^2 \]
Thus,
\[ \sqrt{28 + 6\sqrt{3}} = 3\sqrt{3} + 1 \]

### Step 6: Combine the results
Now we can combine the two results:
\[ \sqrt{28 - 6\sqrt{3}} + \sqrt{28 + 6\sqrt{3}} = (3\sqrt{3} - 1) + (3\sqrt{3} + 1) \]
This simplifies to:
\[ 3\sqrt{3} - 1 + 3\sqrt{3} + 1 = 6\sqrt{3} \]

### Final Answer
Thus, the value of \( \sqrt{28 - 6\sqrt{3}} + \sqrt{28 + 6\sqrt{3}} \) is:
\[ \boxed{6\sqrt{3}} \]