SQUARES AND SQUARE ROOTS A REVIEW. CONTENTS SQUARES. PERFECT SQUARES. FACTS ABOUT SQUARES. SOME METHODS TO FINDING SQUARES. SOME IMPORTANT PATTERNS.

SQUARES AND SQUARE ROOTSA REVIEW

WELCOMETO POWERPOINT

PRESENTATION

TOPICSQUARES AND SQUARE

ROOTS

CONTENTS

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

SQUARES

If a whole number is multiplied by itself, the product is called the square of that number.

For Examples: 1 x 1 = 1 = 12

The square of 1 is 1. 2 x 2 = 4 = 22

The square of 2 is 4

2

2

1

1

3 x 3 = 9 = 32

4 x 4 = 16 = 42

3

3

4

4

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

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.

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

(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.

DIAGONAL METHOD FOR SQUARING

Example:- Find (72)2 using the diagonal method.SOLUTION:-

Therefore, (72)2 =5184.‘FIND (23)2’

ALTERNATIVE METHOD

ALTERNATIVE METHOD

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 = ------- = -------

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

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

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)

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.

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.

Write a Pythagorean Triplet whose one member is 6.

EVALUATION

• EXCEL QUIZ

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