Which of the following numbers is the exact square root of 14161?

Introduction / Context:
This problem checks the ability to identify the square root of a given perfect square without using a calculator. Such questions are standard in aptitude exams, where speed and estimation are very important.

Given Data / Assumptions:

• We are given the number 14161.
• Options are 119, 121, 131, and 141.
• We assume that one of these numbers is the exact positive square root of 14161.

Concept / Approach:
To find the square root of 14161, we can estimate based on nearby squares. Notice that 120^2 = 14400, which is slightly larger than 14161. So the square root must be slightly less than 120. The most reasonable option near 120 is 119, so we explicitly compute 119^2 to check if it equals 14161.

Step-by-Step Solution:
Step 1: Observe that 120^2 = 14400, which is greater than 14161.Step 2: This suggests that the square root must be less than 120.Step 3: Among the options, 119 is closest to 120 from below.Step 4: Compute 119^2. We can use (120 - 1)^2 = 120^2 - 2*120*1 + 1^2.Step 5: 120^2 = 14400, 2*120*1 = 240, and 1^2 = 1.Step 6: So 119^2 = 14400 - 240 + 1 = 14400 - 239 = 14161.Step 7: Since 119^2 equals 14161 exactly, the square root of 14161 is 119.

Verification / Alternative check:
We can also check another option to see that it fails. For example, 121^2 = (120 + 1)^2 = 14400 + 240 + 1 = 14641, which is larger than 14161. This confirms that 121 is not the correct square root.

Why Other Options Are Wrong:

• 121: Its square is 14641, which is greater than 14161.
• 131: Since it is larger than 120, its square will be significantly larger than 14400, clearly exceeding 14161.
• 141: This is also far larger than 120, and 141^2 would be well above the target number.

Common Pitfalls:
Some students may guess based purely on the last digit of the number or may choose 121 because they know it is 11^2 multiplied by 11, but that pattern does not match 14161. Careful estimation around 120 and exact computation avoids these mistakes.

Final Answer:
The exact square root of 14161 is 119.