Which of the following numbers is the exact square root of 15376?

Introduction / Context:
This question again checks mental computation of square roots of perfect squares. Being able to quickly identify square roots near familiar benchmark values such as 100 or 150 is useful in competitive exams.

Given Data / Assumptions:

• We are given the number 15376.
• Options: 122, 124, 134, 128.
• We need the positive integer whose square is exactly 15376.

Concept / Approach:
First, estimate the rough size. Since 120^2 = 14400 and 130^2 = 16900, the square root of 15376 must lie between 120 and 130. We can test the values close to the middle, such as 124 or 128, using algebraic identities to avoid full multiplication.

Step-by-Step Solution:
Step 1: Note that 120^2 = 14400 and 130^2 = 16900, so the square root is between 120 and 130.Step 2: Check 124, which is slightly closer to the lower end.Step 3: Use the identity (a + b)^2 = a^2 + 2ab + b^2 with a = 120 and b = 4.Step 4: Compute 120^2 = 14400.Step 5: Compute 2ab = 2*120*4 = 960.Step 6: Compute b^2 = 4^2 = 16.Step 7: Add them: 14400 + 960 + 16 = 15376.Step 8: Since 124^2 = 15376, the square root of 15376 is 124.

Verification / Alternative check:
We can quickly check another nearby option, such as 122. Using (120 + 2)^2 = 14400 + 480 + 4 = 14884, which is smaller than 15376. That confirms the root must be larger than 122, and our computed value 124 fits correctly.

Why Other Options Are Wrong:

• 122: 122^2 = 14884, which is less than 15376.
• 128: 128^2 = (120 + 8)^2 = 14400 + 1920 + 64 = 16384, larger than 15376.
• 134: 134^2 is even larger than 16900, far above 15376.

Common Pitfalls:
Some test takers try to directly multiply large numbers without using identities, which is slow and error prone. Others may choose values based only on rough estimation without checking the exact square. Using (a + b)^2 helps compute squares efficiently and accurately.

Final Answer:
The exact square root of 15376 is 124.