Find the value of `sqrt(4 + sqrt(44 + sqrt(10000)))`

Options

A. 12
B. 8
C. 4
D. -4

Answer

To find the value of \( \sqrt{4 + \sqrt{44 + \sqrt{10000}}} \), we can break it down step by step.

### Step 1: Simplify the innermost square root
First, we need to calculate \( \sqrt{10000} \):
\[ \sqrt{10000} = 100 \]

### Step 2: Substitute back into the expression
Now, substitute \( 100 \) back into the expression:
\[ \sqrt{4 + \sqrt{44 + 100}} \]

### Step 3: Simplify the next square root
Next, calculate \( \sqrt{44 + 100} \):
\[ 44 + 100 = 144 \]
So, we have:
\[ \sqrt{144} = 12 \]

### Step 4: Substitute back into the expression
Now substitute \( 12 \) back into the expression:
\[ \sqrt{4 + 12} \]

### Step 5: Simplify the final square root
Now calculate \( 4 + 12 \):
\[ 4 + 12 = 16 \]
So, we have:
\[ \sqrt{16} = 4 \]

### Final Answer
Thus, the value of \( \sqrt{4 + \sqrt{44 + \sqrt{10000}}} \) is:
\[ \boxed{4} \]