`root(3)(15612 + sqrt(154 + sqrt(225)))` is equal to

Options

A. 15
B. 25
C. 75
D. 125

Answer

To solve the expression \( \sqrt[3]{15612 + \sqrt{154 + \sqrt{225}}} \), we will break it down step by step.

### Step 1: Simplify \( \sqrt{225} \)
First, we calculate \( \sqrt{225} \):
\[ \sqrt{225} = 15 \]

### Step 2: Substitute \( \sqrt{225} \) back into the expression
Now, we substitute \( \sqrt{225} \) back into the expression:
\[ \sqrt{154 + \sqrt{225}} = \sqrt{154 + 15} \]

### Step 3: Add \( 154 \) and \( 15 \)
Next, we add \( 154 \) and \( 15 \):
\[ 154 + 15 = 169 \]

### Step 4: Simplify \( \sqrt{169} \)
Now, we calculate \( \sqrt{169} \):
\[ \sqrt{169} = 13 \]

### Step 5: Substitute \( \sqrt{169} \) back into the expression
Now, we substitute \( \sqrt{169} \) back into the expression:
\[ \sqrt[3]{15612 + \sqrt{154 + \sqrt{225}}} = \sqrt[3]{15612 + 13} \]

### Step 6: Add \( 15612 \) and \( 13 \)
Next, we add \( 15612 \) and \( 13 \):
\[ 15612 + 13 = 15625 \]

### Step 7: Calculate \( \sqrt[3]{15625} \)
Finally, we calculate the cube root:
\[ \sqrt[3]{15625} = 25 \]

### Final Answer
Thus, the value of \( \sqrt[3]{15612 + \sqrt{154 + \sqrt{225}}} \) is \( 25 \).

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