What is the square root of 1485961?

Introduction / Context:
This question checks the ability to handle large perfect squares. Instead of performing long multiplication from scratch, using algebraic identities around a nearby convenient base, such as 1200, helps in efficient computation.

Given Data / Assumptions:

• We must find the square root of 1485961.
• Options: 1213, 1219, 1229, 1239.
• One of these is the exact square root.

Concept / Approach:
Since 1200^2 = 1440000, our target number 1485961 is slightly above this. So the square root will be somewhat greater than 1200. Among the options, 1219, 1229, and 1239 are all close to 1200, but we can test 1219 first, using (1200 + 19)^2 or (1220 - 1)^2 to compute efficiently.

Step-by-Step Solution:
Step 1: Consider 1219 as a candidate.Step 2: Write 1219^2 as (1200 + 19)^2 = 1200^2 + 2*1200*19 + 19^2.Step 3: Compute 1200^2 = 1440000.Step 4: Compute 2*1200*19 = 2400*19 = 45600.Step 5: Compute 19^2 = 361.Step 6: Add: 1440000 + 45600 = 1485600.Step 7: Add 361 to get 1485961.Step 8: Therefore 1219^2 = 1485961 exactly, so the square root is 1219.

Verification / Alternative check:
To confirm, we can quickly test another nearby option, such as 1213 or 1229, and see that their squares do not match 1485961. This provides additional confidence that 1219 is the unique correct choice among the options.

Why Other Options Are Wrong:

• 1213: Its square will differ from 1485961 because changing the base from 1219 to 1213 changes the value by a significant margin.
• 1229: This is larger than 1219, and its square will exceed 1485961.
• 1239: This is even larger, yielding a much greater square.

Common Pitfalls:
Students may attempt direct multiplication without using identities, which is slow and error prone for such large numbers. Another mistake is estimating based on too wide an interval instead of testing the most reasonable candidate precisely.

Final Answer:
The square root of 1485961 is 1219.