Evaluate:
`sqrt (14161)`

Options

A. `149`
B. `109`
C. `119`
D. `129`

Answer

To evaluate the square root of 14161, we can follow these steps:

### Step 1: Identify the number
We need to find the square root of 14161.

### Step 2: Prime Factorization
We will factor 14161 into its prime factors.

To do this, we can start dividing by the smallest prime numbers:
- 14161 is not divisible by 2 (since it's odd).
- Check divisibility by 3: The sum of the digits (1 + 4 + 1 + 6 + 1 = 13) is not divisible by 3.
- Check divisibility by 5: It does not end in 0 or 5.
- Check divisibility by 7:
- 14161 ÷ 7 = 2023 (exact division, so 7 is a factor).

Next, we factor 2023:
- 2023 ÷ 7 = 289 (exact division, so 7 is a factor again).

Now, we factor 289:
- 289 = 17 × 17 (since 17 is a prime number).

Thus, we can express 14161 as:
\[ 14161 = 7 \times 7 \times 17 \times 17 \]

### Step 3: Simplifying the Square Root
Now, we can take the square root of 14161:
\[ \sqrt{14161} = \sqrt{(7 \times 7) \times (17 \times 17)} \]
Using the property of square roots:
\[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]
This gives us:
\[ \sqrt{14161} = \sqrt{7^2} \times \sqrt{17^2} = 7 \times 17 \]

### Step 4: Calculate the Final Value
Now we calculate \( 7 \times 17 \):
\[ 7 \times 17 = 119 \]

### Conclusion
Thus, the square root of 14161 is:
\[ \sqrt{14161} = 119 \]

### Final Answer
The answer is **119**.

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