Find the square root `7+sqrt48`.

Options

A. `2-sqrt3`
B. `2+sqrt3`
C. `3+sqrt2`
D. `3-sqrt2`

Answer

To find the square root of \( 7 + \sqrt{48} \), we will follow these steps:

### Step 1: Simplify \( \sqrt{48} \)
First, we need to simplify \( \sqrt{48} \). We can factor \( 48 \) into its prime factors:
\[ 48 = 16 \times 3 = 4^2 \times 3 \]
Thus, we can write:
\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression:
\[ 7 + \sqrt{48} = 7 + 4\sqrt{3} \]

### Step 3: Recognize the perfect square
Next, we will express \( 7 + 4\sqrt{3} \) in the form of a perfect square. We can observe that:
\[ 7 + 4\sqrt{3} = (2 + \sqrt{3})^2 \]
To verify:
\[ (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Step 4: Take the square root
Now we can take the square root of the expression:
\[ \sqrt{7 + 4\sqrt{3}} = \sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3} \]

### Final Answer
Thus, the square root of \( 7 + \sqrt{48} \) is:
\[ \boxed{2 + \sqrt{3}} \]