find`sqrt(7+sqrt(48))`=______

Answer

To solve the problem \( \sqrt{7 + \sqrt{48}} \), we can follow these steps:

### Step 1: Simplify \( \sqrt{48} \)
First, we need to simplify \( \sqrt{48} \):
\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]
So, we can rewrite the expression:
\[ \sqrt{7 + \sqrt{48}} = \sqrt{7 + 4\sqrt{3}} \]

### Step 2: Rewrite 7
Next, we can express 7 in a form that will help us use the identity \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ 7 = 4 + 3 \]
Thus, we can rewrite the expression:
\[ \sqrt{7 + 4\sqrt{3}} = \sqrt{(4 + 3) + 4\sqrt{3}} = \sqrt{4 + 3 + 4\sqrt{3}} \]

### Step 3: Identify \( a \) and \( b \)
Now, we can identify \( a \) and \( b \) for the identity:
Let \( a = \sqrt{4} = 2 \) and \( b = \sqrt{3} \).

### Step 4: Use the identity
We can now express \( 7 + 4\sqrt{3} \) as:
\[ (2 + \sqrt{3})^2 = 2^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]
Thus, we have:
\[ \sqrt{7 + 4\sqrt{3}} = \sqrt{(2 + \sqrt{3})^2} \]

### Step 5: Simplify the square root
Taking the square root gives us:
\[ \sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3} \]

### Final Answer
Thus, the final answer is:
\[ \sqrt{7 + \sqrt{48}} = 2 + \sqrt{3} \]

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