`sqrt(8 + sqrt(57 + sqrt(38 + sqrt(108 + sqrt(169)))))=? `

Options

A. 4
B. 6
C. 8
D. 10

Answer

To solve the expression \( \sqrt{8 + \sqrt{57 + \sqrt{38 + \sqrt{108 + \sqrt{169}}}}} \), we will simplify it step by step.

### Step 1: Simplify the innermost square root
First, we start with the innermost expression \( \sqrt{169} \):
\[ \sqrt{169} = 13 \]

### Step 2: Substitute back into the expression
Now substitute \( 13 \) back into the expression:
\[ \sqrt{108 + \sqrt{169}} = \sqrt{108 + 13} = \sqrt{121} \]

### Step 3: Simplify the next square root
Now simplify \( \sqrt{121} \):
\[ \sqrt{121} = 11 \]

### Step 4: Substitute back again
Now substitute \( 11 \) back into the expression:
\[ \sqrt{38 + \sqrt{108 + \sqrt{169}}} = \sqrt{38 + 11} = \sqrt{49} \]

### Step 5: Simplify \( \sqrt{49} \)
Now simplify \( \sqrt{49} \):
\[ \sqrt{49} = 7 \]

### Step 6: Substitute back into the expression
Now substitute \( 7 \) back into the expression:
\[ \sqrt{57 + \sqrt{38 + \sqrt{108 + \sqrt{169}}}} = \sqrt{57 + 7} = \sqrt{64} \]

### Step 7: Simplify \( \sqrt{64} \)
Now simplify \( \sqrt{64} \):
\[ \sqrt{64} = 8 \]

### Step 8: Substitute back into the outermost expression
Now substitute \( 8 \) back into the expression:
\[ \sqrt{8 + \sqrt{57 + \sqrt{38 + \sqrt{108 + \sqrt{169}}}}} = \sqrt{8 + 8} = \sqrt{16} \]

### Step 9: Final simplification
Finally, simplify \( \sqrt{16} \):
\[ \sqrt{16} = 4 \]

Thus, the final answer is:
\[ \sqrt{8 + \sqrt{57 + \sqrt{38 + \sqrt{108 + \sqrt{169}}}}} = 4 \]