What is the value of `sqrt(7 + 4sqrt(3))?`

Options

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

Answer

To find the value of \( \sqrt{7 + 4\sqrt{3}} \), we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \sqrt{7 + 4\sqrt{3}} \]

### Step 2: Rewrite the expression
We can rewrite \( 7 \) as \( 4 + 3 \):
\[ \sqrt{4 + 3 + 4\sqrt{3}} \]

### Step 3: Recognize the form of a perfect square
Notice that \( 4 + 3 + 4\sqrt{3} \) can be expressed in the form \( (a + b)^2 \). We can try to express it as:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
where \( a^2 = 4 \) and \( b^2 = 3 \).

### Step 4: Determine \( a \) and \( b \)
From \( a^2 = 4 \), we find \( a = 2 \).
From \( b^2 = 3 \), we find \( b = \sqrt{3} \).
Now, we calculate \( 2ab \):
\[ 2ab = 2 \cdot 2 \cdot \sqrt{3} = 4\sqrt{3} \]

### Step 5: Combine the results
Thus, we can write:
\[ 7 + 4\sqrt{3} = (2 + \sqrt{3})^2 \]

### Step 6: Take the square root
Now, we take the square root of both sides:
\[ \sqrt{7 + 4\sqrt{3}} = \sqrt{(2 + \sqrt{3})^2} \]

### Step 7: Simplify
This simplifies to:
\[ 2 + \sqrt{3} \]

### Final Answer
Thus, the value of \( \sqrt{7 + 4\sqrt{3}} \) is:
\[ \boxed{2 + \sqrt{3}} \]

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