The value of `sqrt(176+sqrt(2401))` is

Options

A. `14`
B. `15`
C. `16`
D. `17`

Answer

To solve the expression \( \sqrt{176 + \sqrt{2401}} \), we will follow these steps:

### Step 1: Calculate \( \sqrt{2401} \)
First, we need to find the square root of 2401.

To do this, we can factor 2401:
- 2401 can be divided by 7:
- \( 2401 \div 7 = 343 \)
- 343 can again be divided by 7:
- \( 343 \div 7 = 49 \)
- 49 can be divided by 7:
- \( 49 \div 7 = 7 \)
- Finally, 7 can be divided by 7:
- \( 7 \div 7 = 1 \)

Thus, we have:
\[ 2401 = 7^4 \]

Taking the square root:
\[ \sqrt{2401} = \sqrt{7^4} = 7^2 = 49 \]

### Step 2: Substitute \( \sqrt{2401} \) back into the expression
Now we substitute \( \sqrt{2401} \) back into the original expression:
\[ \sqrt{176 + \sqrt{2401}} = \sqrt{176 + 49} \]

### Step 3: Calculate \( 176 + 49 \)
Now we add 176 and 49:
\[ 176 + 49 = 225 \]

### Step 4: Calculate \( \sqrt{225} \)
Next, we need to find the square root of 225:
- We can factor 225:
- \( 225 = 15 \times 15 \)

Thus:
\[ \sqrt{225} = 15 \]

### Final Answer
Therefore, the value of \( \sqrt{176 + \sqrt{2401}} \) is \( 15 \).

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