Square numbers can only end with which digits?

1, 4, 5, 6, 9, or 0

Fill in the blank: The square root of 5184 is ___.

72

True or False: The number 4539 can be a perfect square.

False; the digit sum of 3 (4 + 5 + 3 + 9 = 21, 2 + 1 = 3) is not 1, 4, 7, or 9.

How do you square a number ending in 5, like 95²?

Multiply 9 by 10 (9 + 1) to get 90, then add 25 (5²). Thus, 95² = 9025.

Can a number with a digit sum of 2 be a perfect square?

No, a perfect square’s digit sum must be 1, 4, 7, or 9.

What is the duplex of 45 in the duplex method?

For two digits, duplex is 2 × (4 × 5) = 40. Thus, D(45) = 40.

How do you square a number above 50, such as 54²?

 First part: 5² + 4 = 25 + 4 = 29. 

Second part: 4² = 16. Thus, 54² = 2916.

Calculate 1003² using the near-base method (base 1000).

Deviation is +3. 
First part: 1003 + 3 = 1006.
 Second part: 3² = 009 (three digits). 
Thus, 1003² = 1006009.

Which operation is performed after finding the first digit of the square root?

Subtract the square of the first digit from the left pair to find the remainder.

 Explain why the number 87654 is not a perfect square.

87654 ends in 4 (valid) but has a digit sum of 3 (8 + 7 + 6 + 5 + 4 = 30, 3 + 0 = 3), which is not 1, 4, 7, or 9, so it cannot be a perfect square.

How do you determine the remainder when calculating square roots?

Subtract the square of the current digit from the current number to find the remainder.