`(sqrt(10+sqrt(25+sqrt(108+sqrt(154+sqrt(225))))))/(root3(8))=?`

Options

A. `4`
B. `2`
C. `8`
D. `(1)/(2)`

Answer

To solve the expression \(\frac{\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}}}{\sqrt[3]{8}}\), we will break it down step by step.

### Step 1: Simplify the denominator
The denominator is \(\sqrt[3]{8}\). We know that:
\[ \sqrt[3]{8} = 2 \]

### Step 2: Simplify the innermost square root
Now, let's simplify the expression in the numerator starting from the innermost square root:
\[ \sqrt{225} = 15 \]
So, we replace \(\sqrt{225}\) in the expression:
\[ \sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + 15}}}} \]

### Step 3: Simplify the next square root
Next, we simplify \(\sqrt{154 + 15}\):
\[ 154 + 15 = 169 \quad \text{and} \quad \sqrt{169} = 13 \]
Now, we have:
\[ \sqrt{10 + \sqrt{25 + \sqrt{108 + 13}}} \]

### Step 4: Simplify the next square root
Next, we simplify \(\sqrt{108 + 13}\):
\[ 108 + 13 = 121 \quad \text{and} \quad \sqrt{121} = 11 \]
Now, we have:
\[ \sqrt{10 + \sqrt{25 + 11}} \]

### Step 5: Simplify the next square root
Next, we simplify \(\sqrt{25 + 11}\):
\[ 25 + 11 = 36 \quad \text{and} \quad \sqrt{36} = 6 \]
Now, we have:
\[ \sqrt{10 + 6} \]

### Step 6: Simplify the final square root
Now, we simplify \(\sqrt{10 + 6}\):
\[ 10 + 6 = 16 \quad \text{and} \quad \sqrt{16} = 4 \]

### Step 7: Combine the results
Now we can substitute back into our original expression:
\[ \frac{\sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}}}{\sqrt[3]{8}} = \frac{4}{2} = 2 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{2} \]