Answer:
1. (you could have used your calculator too ...)
625
46225
11.56
1.5625
0.3364
2. a, b, c, d are not prefect squares
3. a and c squares will end with a "1" in the units place.
4. b and c squares will end with a "6" in the units place.
5. a. 17
b. 161
6. 25
7. a. 144 + 145
b. 264 + 265
c. 480 + 481
d. 1012 + 1013
8. a, d, e
9. a. d
Step-by-step explanation:
1. is just plain calculator work
2. a. no perfect square
Property 1: A number having 2, 3, 7 or 8 at unit’s place is never a perfect square. In other words, no square number ends in 2, 3, 7 or 8.
2. b. no perfect square
Property 2: The number of zeros at the end of a perfect square is always even. In other words, a number ending in an odd number of zeros is never a perfect square.
2. c. no perfect square
the same as 2.b.
2. d. no perfect square
the same as 2.a.
2. e. is a perfect square (of 250)
3.
Property 6: The unit’s digit of the square of a natural number is the unit’s digit of the square of the digit at unit’s place of the given natural number.
for 81² : as 1²=1, therefore 81² will also end with a 1.
for 119² : as 9²=81 ends with 1, 119² will also end with a 1.
all others do not meet that criteria.
4.
again Property 6.
44² : 4²=16, therefore 44² will also end with a 6.
126² : 6² = 36, therefore 126² will also end with a 6.
5.
given a number n and its following number n+1, the difference between their squares is
(n+1)² - n² = n² + 2n + 1 - n² = 2n + 1
there cannot be any perfect square number in between n² and (n+1)², as the base number would have to be between n and n+1 (which is impossible).
a. 2×8 + 1 = 17
b. 2×80 + 1 = 161
6. as 5.
2×12 + 1 = 25
7.a.
n + n + 1 = 17²
2n + 1 = 17²
n = (17² - 1)/2 = 144
n+1 = 145
b.
2n + 1 = 23²
n = (23² - 1)/2 = 264
n+1 = 265
c.
n = (31² - 1)/2 = 480
n+1 = 481
d.
n = (45² - 1)/2 = 1012
n+1 = 1013
8.
Property 3: Squares of even numbers are always even numbers and square of odd numbers are always odd.
therefore, only 256, 1296 and 144 can be squares of even numbers.
9.
again Property 3.
therefore, only 529 and 361 can be squares of odd numbers.
