To find the least number that must be added to 6412 to make it a perfect square, follow these steps:

1. Estimate the nearest square root of 6412: Calculate the approximate square root of 6412.

2. Identify the nearest integers around the square root: Find the nearest whole numbers less than and greater than the square root of 6412.

3. Calculate the squares of these integers: Compute the squares of these two integers to determine the closest perfect squares above and below 6412.

4. Subtract 6412 from the nearest higher perfect square: The difference between 6412 and the nearest higher perfect square is the least number that must be added.

Let’s calculate these steps:

1. Approximate Square Root: $\sqrt{6412} \approx 80.1$

2. Nearest Whole Numbers: The integers around 80.1 are 80 and 81.

3. Perfect Squares:

   • $80^2 = 6400$
   • $81^2 = 6561$

4. Find the Difference:

   • $6561 - 6412 = 149$

Therefore, the least number that must be added to 6412 to make it a perfect square is 149.

Would you like further details or have any questions?

Here are some related questions:

1. What is a perfect square and how is it defined?
2. How do you find the nearest perfect square of a number?
3. What is the process to determine whether a number is a perfect square?
4. How can you calculate square roots without a calculator?
5. What other ways can be used to approximate square roots?

Tip: Always double-check calculations when dealing with approximations or manual square root estimations to ensure accuracy!