Find the exact value of 1397 multiplied by 1397, that is, compute 1397 * 1397 without using a calculator.

Introduction / Context:
This problem asks for the square of a four digit number, 1397. Although it looks like a direct multiplication question, it also tests your familiarity with algebraic identities and your ability to simplify calculations using nearby convenient numbers. Such questions appear in mental math and simplification sections, where using identities instead of raw multiplication can save significant time.

Given Data / Assumptions:

• We need to compute 1397 * 1397.
• All calculations are in standard decimal arithmetic.
• No calculator is to be used in typical exam conditions.

Concept / Approach:
Instead of directly multiplying 1397 by 1397 using long multiplication, we can view 1397 as a number close to 1400. Then we can apply the identity (a - b)^2 = a^2 - 2ab + b^2. Here, a = 1400 and b = 3, because 1397 = 1400 - 3. This method makes the squaring much easier because squaring 1400 and computing the remaining terms is straightforward when compared to full long multiplication.

Step-by-Step Solution:
Write 1397 as 1400 - 3. We need (1400 - 3)^2. Use the identity (a - b)^2 = a^2 - 2ab + b^2. Here, a = 1400 and b = 3. Compute a^2 = 1400^2 = 1400 * 1400. 1400 * 1400 = 14 * 14 * 100 * 100 = 196 * 10000 = 1960000. Compute 2ab = 2 * 1400 * 3 = 2800 * 3 = 8400. Compute b^2 = 3^2 = 9. Now, (1400 - 3)^2 = 1960000 - 8400 + 9. First subtract: 1960000 - 8400 = 1951600. Then add 9: 1951600 + 9 = 1951609. Therefore, 1397 * 1397 = 1951609.

Verification / Alternative check:
A long multiplication method would also confirm the same result, though it is more laborious. Breaking 1397 into 1400 - 3 and squaring via the identity reduces the risk of arithmetic mistakes. Additionally, a quick digital sum check (casting out nines) can be used. The sum of digits of 1397 is 1 + 3 + 9 + 7 = 20, and 20 reduces to 2 + 0 = 2. The square of a number whose digit sum is 2 has digit sum 4 (since 2^2 = 4). The digits of 1951609 sum to 1 + 9 + 5 + 1 + 6 + 0 + 9 = 31, which reduces to 3 + 1 = 4, consistent with the expected check, supporting the correctness of 1951609.

Why Other Options Are Wrong:
The other options differ significantly from 1951609 and would arise from mistakes such as forgetting to add b^2, miscalculating 2ab, or mixing up intermediate subtraction steps. For instance, using 1960000 - 8300 instead of 8400 or misplacing digits when adding the final 9 could lead to some of these incorrect values.

Common Pitfalls:
The main pitfall is attempting full long multiplication under time pressure and making digit alignment errors. Another common error is to misremember the identity as a^2 + 2ab + b^2 instead of a^2 - 2ab + b^2 for a minus sign, which gives a completely different result. Also, while simplifying 1960000 - 8400, it is easy to subtract incorrectly. Using neat stepwise calculation and checking intermediate steps helps avoid these issues.

Final Answer:
The exact value of 1397 multiplied by 1397 is 1951609.