This document provides an overview of squares and square roots including: - The definition of a square as a number multiplied by itself - Perfect squares being numbers where the square root is also a whole number - Patterns involving squares such as even/odd properties and formulas for decomposing squares - Methods for finding squares including the diagonal method and using formulas like (a + b)2 - Pythagorean triplets being sets of three numbers that satisfy the Pythagorean theorem when used as sides of a right triangle

1. SQUARES AND SQUARE ROOTS
A REVIEW

5. CONTENTS
• SQUARES.
• PERFECT SQUARES.
• FACTS ABOUT SQUARES.
• SOME METHODS TO FINDING SQUARES.
• SOME IMPORTANT PATTERNS.
• PYTHAGOREAN TRIPLET.

6. SQUARES
If a whole number is multiplied by
itself, the product is called the square
1
of that number.
For Examples: 1 x 1 = 1 = 12 1
The square of 1 is 1. 2
2 x 2 = 4 = 22
2
The square of 2 is 4

7. 3
3
3x3=9= 32
4
4 x 4 = 16 = 42
4

8. PERFECT SQUARE
A natural number ‘x’ is a perfect
square, if y2 = x where ‘y’ is natural
number. Examples : 16 and 25 are
perfect squares, since
16 = 42
25 = 52

9. FACTS ABOUT SQUARES
• A number ending with 2, 3, 7 or 8 is never a
perfect square.
• The squares of even numbers are even.
• The squares of odd numbers are odd.
• A number ending with an odd number of zeros is
never a perfect square.
• The ending digits of a square number is 0, 1, 4, 5,
6 or 9 only.
Note : it is not necessary that all numbers ending
with digits 0, 1, 4, 5, 6 or 9 are square numbers.

10. SOME METHODS TO FINDING
SQUARES
USING THE FORMULA
( a + b )2 = a2 + 2ab + b2
1. (27)2 = (20 + 7 )2
(20 + 7)2 = (20)2 + 2 x 20x 7 + (7)2
= 400 + 280 + 49 = 729
. FIND (32)2

11. (a – b )2 = a2 – 2ab + b2
1. (39)2 = (40 -1)2
(40 – 1)2 = (40)2 – 2 x 40 x 1 + (1)2
= 1600 – 80 + 1 = 1521
. FIND (48)2.

12. DIAGONAL METHOD FOR
SQUARING
Example:- Find (72)2 using the
diagonal method.
SOLUTION:-

13. Therefore,
(72)2 =5184.
‘FIND (23)2’

14. ALTERNATIVE METHOD
ALTERNATIVE METHOD

15. SOME INTERESTING PATTERNS
1. SQUARES ARE SUM OF CONSECUTIVE
ODD NUMBERS.
EXAMPLES:
1 + 3 = 4 = 22
1 + 3 + 5 = 9 = 32
1+3+5+7 = 16 = 42
1+3+5+7+9 = 25 = 52
1+3+5+7+9+11 = ------- = -------

16. 2. SQUARES OF NUMBERS ENDING WITH
DIGIT 5.
(15)2 =1X (1 + 1)X 100 +25
= 1X2X100 + 25
= 200 + 25 = 225
(25)2 = 2X3X100 + 25
= 600 + 25 = 625
(35)2 = (3X4) 25 = 1225
TENS UNITS
FIND (45)2

17. PYTHAGOREAN TRIPLETS
If three numbers x, y and z are such that
x2 + y2 = z2, then they are called
Pythagorean Triplets and they represent
the sides of a right triangle.
x z
y

18. Examples
(i) 3, 4 and 5 form a Pythagorean
Triplet. 32 + 42 = 52.( 9 + 16 = 25)
(ii) 8, 15 and 17 form a Pythagorean
Triplet. 82+152 = 172.
(64 +225 = 289)

19. Find Pythagorean Triplet if one
element of a Pythagorean Triplet
is given.
For any natural number n, (n>1), we have
(2n)2 + (n2-1)2 = (n2+1)2.
such that 2n, n2-1 and n2+1 are Pythagorean
Triplet.

20. Examples-
Write a Pythagorean Triplet whose
one member is 12.
Since, Pythagorean Triplet are 2n,
n2-1 and n2+1.
So, 2n = 12, n = 6.
n2-1 = (6)2-1 = 36 -1= 35
And n2+1 = (6)2+1= 36+1= 37
Therefore, 12, 35 and 37 are Triplet.

21. Write a Pythagorean Triplet
whose one member is 6.

22. EVALUATION
• EXCEL QUIZ

23. My sincere thanks to :- NVS R. O.
Bhopal and our Principal Ms. Kavita
Singh for providing me an
opportunity to prepare a PPT, and
also to Mrs. Anju Pandey, TGT Eng,
For her cooperation.