Squares are numbers obtained by multiplying an integer by itself. Perfect squares are integers whose square root is also an integer, such as 4, 9, and 100. The properties of squares include that numbers ending in 2, 3, 7, or 8 cannot be perfect squares and that the number of zeros at the end of perfect squares is always even. Square roots are values that when multiplied by themselves equal the original number. Common methods for finding square roots include division, prime factorization, and repeated subtraction. Square roots and squares also follow properties such as squares of even numbers being even and odd numbers being odd.

1. TOPIC:- SQUARES AND SQUARE
ROOT

2. CONTENTS
 Squares
 Square root

3. SQUARES
 The number you get when you multiply an
integer by itself.
Example: 4 × 4 = 16, so 16 is a square number.
 Some Squares number are :-
0 (=0×0)
1 (=1×1)
4 (=2×2)
9 (=3×3)
16 (=4×4)
25 (=5×5

4. PERFECT SQUARE:-
 A perfect square is any integer whose square
root is also an integer.
Example :-
4, 9, 100 , etc.

5. PROPERTES OF SQUARES:-
 A number ending with 2,3,7 and 8 can never be a
perfect square.
 The number of zeros at the end of perfect squares is
always even.
Example:- √10000 = 1002
 Squares of even numbers are even and odd are odd.

6.  For any two consecutive natural number n
and (n+1) we have,
(n+1)2 – n2 = (n+1+n)(n+1-n) = (n+1) + n
 For any number n greater than 1 , the
Pythagorean triplet is given by (2n,n2-1,n2+1)

7. SQUARE ROOT
 The square root of a number is a value
that, when multiplied by itself, gives the
number.
Example: 4 × 4 = 16, so the square root of 16 is
4.
The symbol is √
Another example: √36 = 6 (because 6 x 6 = 36)

8. METHOD TO FIND SQUARE ROOT
 Division method
 Prime factorization method
 Repeated subtraction

9. Division method
 This describes a "long hand" or manual
method of calculating or extracting square
roots. Calculation of a square root by hand is
a little like long-hand division.

10. Prime factorization method
 According to this method we express the
given number as the product of prime factors.
Example :-
√4=2*2=2

11. Repeated subtraction
 Every perfect square number can be
expressed as a sum of successive odd
natural numbers starting from 1.
Example :- √9=9-1=8
=8-3=5
=5-5=0
So ,square root is 3.

12. Properties of Square root
 If the unit digit of a number is 2,3,7,and8 it
does not a have square root.
 If a number ends in an odd number of zeros
then it does not have a square root in N.
 The square root of even is even and odd is
odd.

13. BY:-
NEELABH SHUKLA
Class-8th A
ROLL NO. - 25