Find the value of `sqrt(248 + sqrt(52 + sqrt(144)))`

Options

A. -16
B. `+- 16`
C. 16
D. 16.2

Answer

To solve the expression \( \sqrt{248 + \sqrt{52 + \sqrt{144}}} \), we will break it down step by step.

### Step 1: Simplify the innermost square root
First, we need to simplify \( \sqrt{144} \):
\[ \sqrt{144} = 12 \]

### Step 2: Substitute back into the expression
Now we substitute \( \sqrt{144} \) back into the expression:
\[ \sqrt{52 + \sqrt{144}} = \sqrt{52 + 12} \]
This simplifies to:
\[ \sqrt{52 + 12} = \sqrt{64} \]

### Step 3: Simplify the next square root
Now we simplify \( \sqrt{64} \):
\[ \sqrt{64} = 8 \]

### Step 4: Substitute back into the outer expression
Now we substitute \( \sqrt{64} \) back into the expression:
\[ \sqrt{248 + \sqrt{52 + \sqrt{144}}} = \sqrt{248 + 8} \]
This simplifies to:
\[ \sqrt{248 + 8} = \sqrt{256} \]

### Step 5: Simplify the final square root
Now we simplify \( \sqrt{256} \):
\[ \sqrt{256} = 16 \]

### Conclusion
Thus, the value of \( \sqrt{248 + \sqrt{52 + \sqrt{144}}} \) is:
\[ \boxed{16} \]