The value of `sqrt(5 + sqrt(11 + sqrt(19 + sqrt(29 + sqrt(49)))))` is :

Options

A. 3
B. 9
C. 7
D. 5

Answer

To simplify the expression \( \sqrt{5 + \sqrt{11 + \sqrt{19 + \sqrt{29 + \sqrt{49}}}}} \), we will break it down step by step.

### Step 1: Simplify the innermost square root
First, we start with the innermost square root:
\[ \sqrt{49} = 7 \]
So, we can rewrite the expression as:
\[ \sqrt{5 + \sqrt{11 + \sqrt{19 + \sqrt{29 + 7}}}} \]

### Step 2: Simplify the next square root
Now we simplify \( \sqrt{29 + 7} \):
\[ 29 + 7 = 36 \quad \text{and} \quad \sqrt{36} = 6 \]
Now, we can update our expression:
\[ \sqrt{5 + \sqrt{11 + \sqrt{19 + 6}}} \]

### Step 3: Simplify the next square root
Next, we simplify \( \sqrt{19 + 6} \):
\[ 19 + 6 = 25 \quad \text{and} \quad \sqrt{25} = 5 \]
Now, we have:
\[ \sqrt{5 + \sqrt{11 + 5}} \]

### Step 4: Simplify the next square root
Now we simplify \( \sqrt{11 + 5} \):
\[ 11 + 5 = 16 \quad \text{and} \quad \sqrt{16} = 4 \]
Updating our expression gives:
\[ \sqrt{5 + 4} \]

### Step 5: Final simplification
Finally, we simplify \( \sqrt{5 + 4} \):
\[ 5 + 4 = 9 \quad \text{and} \quad \sqrt{9} = 3 \]

Thus, the value of the original expression is:
\[ \boxed{3} \]