The document discusses various properties of squares and square roots. It defines a square number as the result of multiplying a number by itself. Some key points made include: - Numbers ending in 2, 3, 7, or 8 cannot be perfect squares, while numbers ending in 1, 4, 5, 6, or 9 may be squares - The units digit of a square number depends on the units digit of the original number - Odd squares are odd numbers and even squares are even numbers - A number is not a perfect square if it ends in an odd number of zeros - The Pythagorean theorem relates the squares of the sides of a right triangle - Square roots can be estimated and found using repeated subtraction

1. Class 8
Square and square roots
Prepared by:
Sharda Chauhan
TGT Maths

2. Square Number
The number you get when you multiply by itself.
Example: 4 x 4 = 16, So 16 is the square
number.
0 (= 0 x 0)
4 (= 2 x 2)
9 (= 3 x 3)
16 (= 4 x 4)

3. Squares

4. Square and square root of the number

5. If a number ends with 2, 3, 7 and 8 at unit’s
place. So it must not be a square number.
Example: 132, 72 , 28
If a number ends with 1, 4, 5, 6 and 9 at unit’s
place. So it may be a square number.
Example: 1) 16 (Yes) 16 = 4 x 4
2) 15 (No) As it ends with 5.
Perfect squares

6. Perfect Squares:
3 and 5 are not perfect squares but 9 and 25 are perfect squares.

7. 1) 12
2) 92
1 x 1 = 1 & 9 x 9 = 81
If a number has 1 or 9 in the unit’s place,
Then it’s square ends with 1.
3) 22
4) 82
2 x 2 = 4 & 8 x 8 = 64
If a number has 2 or 8 in the unit’s place,
Then it’s square ends with 4.
Property of unit digit of a square number.

8. 5) 32
6) 72
3 x 3 = 9 & 7 x 7 = 49
If a number has 3 or 7 in the unit’s place,
Then it’s square ends with 9.
7) 42
8) 62
4 x 4 = 16 & 6 x 6 = 36
If a number has 4 or 6 in the unit’s place,
Then it’s square ends with 6.

9. Even and Odd Numbers

10. If a number is odd so it’s square is also odd, if
a number is even so it’s square is also even.
Example:
5) 49 = 72
1) 16 = 42
2) 25 = 52
( Even )
( Odd )
( Odd )
3) 121 = 112 ( Odd )
4) 144 = 122 ( Even )
Even and odd squares

11. If the sum of numbers are in consecutive from 1
and odd so the square root of these numbers is
number of observation.
Example: 1) 1+3+5+7+9+11+13 = 72
2) 1+3+5+7+9+11+13+15+17 = 92
3) 1+3+5+7+9+11+13+15+17+19+21+23 = 122
Adding odd numbers

12. Odd number of zeros
If a number ends with odd number of zeros then
it is not a perfect square.
Example: 1) 30
2) 5000
3) 400000
4) 100
5) 60000
= Not perfect square
= Not perfect square
= Not perfect square
= Perfect square
= Perfect square

13. If a number contain 1 zero at the end, so 2 zeros
will have its square. It means the number
contains zeros will double zeros of its square.
Example: 1) 302
2) 50002
3) 4002
4) 100002
5) 6000002
= 2 zeros
= 6 zeros
= 4 zeros
= 8 zeros
= 10 zeros
Zeros in the square number

14. Numbers between square
numbers
We can find some numbers between two
consecutive square numbers. By finding the
square numbers of any two number.
Between 12 (=1) and 22 (=4)
there are two non square numbers
Between 22 (=4) and 32 (=9)
there are four non square numbers.
= 2, 3.
= 5, 6, 7, 8, 9.

15. Example:
The non square numbers is 1 less than the
difference of two consecutive squares.
(n - 1) – n
1) 9 (=32) and 16 (=42)
2) 16 (=42) and 25 (=52)
3) 25 (=52) and 36 (=62)
4) 36 (=32) and 49 (=72)
5) 49 (=72) and 64 (=82)
= (16 - 1) – 9 = 6
= (25 - 1) – 16 = 8
= (36 - 1) – 25 = 10
= (49 - 1) – 36 = 12
= (64 - 1) – 49 = 14
Non square numbers

17. Pythagoras Theorem
The square of a numberis equal to the square of
two other numbers. This property is called
Pythagoras property and the three numbers used in
this property are called Pythagoras theorem.
5cm
3cm
4cm
52 = 32 + 42
25 = 9 + 16
25 = 25
Right angle triangle

18. The formula to finding more such triplets.
(2m2) + (m2 -1) = (m2 +1)
Example: 1) 62 + 82 = 102
36 + 64 = 100
100 = 100
So 6, 8, 10 are
Pythagorean triplets.
1) 32 + 42 = 52
9 + 16 = 25
25 = 25
So 3, 4, 5 are
Pythagorean triplets.
Pythagorean Triplets

19. 1) Pythagorean triplet whose smallest member is
12.
Finding Pythagorean triplets
1) 2m = 12
M = 6
2) 62 – 1
36 -1 = 35
3) 62 + 1
36 +1 = 37
Check: 122 + 352 = 372
144 + 1225 = 1369
1369 = 1369
So 12, 35, 37 are
Pythagorean triplets.

20. 2) Pythagorean triplet whose member is 5.
2) 2 x 3 = 6
1) m2 – 1 = 5
m2 = 6
m = 6/2 = 3
3) 32 + 1
9 +1 = 10
Check:
52 + 62 = 102
25 + 36 = 100
100 = 100
So 5, 6, 10 are
Pythagorean triplets.

21. Finding square roots
Repeated Subtraction
The subtraction of odd numbers consequently
to the number. Still we get 0. So the number of
observation we subtract to the number is the
square root of that number.

22. Repeated Subtraction
1) 64 – 1 = 63
2) 63 – 3 = 60
3) 60 – 5 = 55
4) 55 – 7 = 48
5) 48 – 9 = 39
6) 39 – 11 = 28
7) 28 – 13 = 15
8) 15 – 15 = 0
= 82
Find the square root of 64.
So the square root of 64 is
8.

23. Find the square root of 89.
1) 89 – 1 = 88
2) 88 – 3 = 85
3) 85 – 5 = 80
4) 80 – 7 = 73
5) 73 – 9 = 64
6) 64 – 11 = 53
7) 53 – 13 = 40
8) 40 – 15 = 25
9) 25 – 17 = 8
If something is remaining
from the subtraction. So, it
is not the perfect square.

24. Prime factorization
2 256
2 128
2 64
2 32
2 16
2 8
2 4
2 2
1
256 = 2 x 2 x 2 x 2 x
2 x 2 x 2 x 2
= 16
2 x 2 x 2 x 2

25. Property of unit digit of square root
1) 1
2) 9
If a square number has 1 or 9 in the
unit’s place, then it’s square root
ends with 1
If a square number has 2 or 8 in the
unit’s place, then it’s square root
ends with 4
3) 2
4) 8

26. 5) 3 If a square number has 3 or 7 in the
unit’s place, then it’s square root
ends with 9.
6) 7
7) 4
8) 6
If a square number has 4 or 6 in the
unit’s place, Then it’s square root
ends with 6.

27. Square Root of Decimals
23.04
Step1) To find the square root of decimal
numbers put a bar on the internal part. Place the
bar on the decimal part.
23.04

28. Step 2) The left most bar 23 and 42 < 23 < 52. Take
this number as the divisor and get the remainder.
4
4 23.04
16
7

29. Step 3) The remainder is 7. Write the number
under the next bar to the right of this remainder,
to get 704. Double the divisor and enter it with a
blank on its right.
4
4 23.04
16
8_ 704

30. Step 4) We know that 88 x 8 = 704, therefore, the new digit
is 8. Divide and get the remainder.
48
4 23.04
16
88 704
704
0
= 4.8

31. Finding Square by Division Method
625
Step1) To find the square of any number put
two bar on the internal part.
6 25

32. Step 2) Find the largest number whose square is less than or
equal to the number under the extreme left bar ( 22 < 52 < 32 ).
Take this number as the divisor. Divide and get the remainder.
2
2 6 25
4
2

33. Step 3) The remainder is 7. Write the number
under the next bar to the right of this remainder,
to get 225. Double the divisor and enter it with a
blank on its right.
2
2 6 25
4
4_ 225

34. Step 4) We know that 45 x 5 = 225, therefore, the
new digit is 5. Divide and get the remainder.
25
2 6 25
16
45 225
225
0
= 25

35. Estimating Square root
In all such cases we need to estimate the
square root.
125 < 122
We know that 112 <
121 < 125 < 144
So 121 is much closer to 125 than 144, therefore
11 is the estimating square root of 125.

37. Quiz
Q1) Which one of the following number is a
perfect squares:
a) 622
b) 393
c) 5778
d) 625
Answer: d) 625

38. Q2) Check which of the following is not a perfect
square :
a) 81000
b) 8100
c) 900
d) 6250000
Answer: a) 81000

39. Q3) Which of the following perfect square numbers,
is the square of an odd number:
a) 289
b) 400
c) 900
d) 1600
Answer: a) 289

40. Q4) Which of the following perfect square numbers,
is the square of an even number:
a) 361
b) 625
c) 4096
d) 2601
Answer: c) 4096

41. Q5) How many natural numbers lie between squares
of 11 and 12.
a) 22
b) 23
c) 24
d) 25
Answer: a) 22
112 , 122 (144 - 1) - 121 = 22

42. Q6) Find the unit digit of (564)2
a) 2
b) 4
c) 6
d) 8
Answer: c) 6

43. Q7) Find the unit digit of square root of (1521)?
a) 1 or 9
b) 4 or 6
c) 2 or 7
d) 8 or 5
Answer: a) 1 or 9

44. Q8) 1 + 3 + 5 +7 + 9 + 11 + 13 + 15 is a perfect square
of number:
a) 8
b) 7
c) 6
d) 9
Answer: b) 8

45. Q9) What is the formula to find Pythagorean triplets:
a) (m) + (m2-1) = (m2+1)
b) (2m) + (m2-1) = (m2+1)
c) (2m) + (m-1) = (m+1)
d) (2m) + (m2) = (m2+1)
Answer: b) (2m) + (m2-1) = (m2+1)

46. Q10) Can a prime number be perfect square:
GUESS
False !
Q11) If a number is odd so it’s square is also
odd:
True !

47. THANKYOU!