Chapter 3 Square and Square Roots Class VIII

Chapter – 3 Square and Square Roots Class - VIII Mr. S M Rahman

Square of First 40 Natural Numbers No Square 1 1 11 121 21 441 31 961 2 4 12 144 22 484 32 1024 3 9 13 169 23 529 33 1089 4 16 14 196 24 576 34 1156 5 25 15 225 25 625 35 1225 6 36 16 256 26 676 36 1296 7 49 17 289 27 729 37 1369 8 64 18 324 28 784 38 1444 9 81 19 361 29 841 39 1521 10 100 20 400 When a number multiplied by itself we get square of that number. 30 900 40 1600 From the above table we can say that the numbers 1, 4, 9, 16… are square numbers and these are called perfect square numbers.

Properties of Perfect Square Number � � � � In the prime factorization of a given number if all the prime factors occur in pairs then the given number is a perfect square number. A number ending with 2, 3, 7 or 8 can never be a perfect square A number ending with an odd number of zeros can never be a perfect square. Square of even number is even and Square of odd number is odd If a number has 1 or 9 in the unit’s place, then its square ends with 1 If a number has 2 or 8 in the unit’s place, then its square ends with 4 If a number has 3 or 7 in the unit’s place, then its square ends with 9 If a number has 4 or 6 in the unit’s place, then its square ends with 6

Pythagorean triplets The formula for finding Pythagorean triplets is (2 m, m 2 – 1, m 2 +1) # Find Pythagorean triplet whose smallest number is 12 2 m =12 , m=6 m 2 – 1= 62 -1 , = 35 m 2 + 1 = 62 + 1 = 37 So, (12, 35, 37) are Pythagorean triplet

Some Interesting Patters of Square

Column Method: Finding the square of 98 ? a = 9, b= 8 Column II a 2 2 xaxb 81 +15 ----96 2 x 9 x 8 144 +6 ------= 150 96 0 Column III b 2 � 64 � Hence (98)2 = 9604 Marks( with bar) the ones digit of column III and then add the tens place digit(if any) to the column II Marks( with bar) the ones digit of column II and then add the tens place digit(if any) to the column I The number formed by the Underlined digits in each column gives the square of the given 2 -digit number

Diagonal Method : Find the square of 72 4 4+0+4=08 1+9+1= 11 04 +1 ---5 Hence square of 72 is 5184

Short cut Methods for Finding Square � SQUARES OF A NUMBERS ENDING WITH 5 752=75 x 75 5625 Steps : 1. Multiply 5 by 5 and put composite digit 25 on the RHS. 2. Then add 1 to the upper left hand digit 7 to make it 8. 3. Then multiply 8 by the lower left –hand digit 7. Put the result (56) on the LHS. 4. Therefore, the answer comes out to be 5625.

THE SQUARE OF AN ADJACENT NUMBER � 452 = 462 = = = 2025 (known) 452 + (45+46) 2025+91 2116 The square of a number, say 60 is easily known. With the help of this you can find out the square of 59? (60)2 = 3600 (known) (59)2 = 3600 -(60+59)=3600 -119=3481

MENTAL FORMULA FOR FINDING SQUARES § § FROM 112 TO 192 First find the square of 11 using the formula : 112 = 122 = 132 = 142 = 152 = 162 = 172 = 182 = 192 = 11 + 1/12 12 + 2/22 13 + 3/32 14 + 4/42 15 + 5/52 16 + 6/62 17 + 7/72 18 + 8/82 19+ 9/92 = 12/1 = 121 = 14/4 = 144 = 16/9 = 169 = 18/16 = 1(8+1)6 = 20/25 = 2(0+2)5 = 22/36 = 2(2+3)6 = 24/49 = 2(4+4)9 = 26/64 = 2(6+6)4 = 28/81 = 2(8+8)1 = = = 196 225 256 289 324 361

FROM 212 TO 292: � 212=2 x (21+1)/12 222=2 x (22+2)/22 232 =2 x (23+3)/32 242 =2 x (24+4)/42 252 =2 x (25+5)/52 262 =2 x (26+6)/62 272 =2 x (27+7)/72 282 = 2 x (28+8)/82 292 =2 x (29+9)/9 =2 x (22)/1 = 441 =2 x(24)/4 = 484 =2 x(26)/9 = 529 =2 x(28)/16 = 56/16 =576 =2 x 30/25 = 625 =2 x 32/36 = 676 =2 x 34/49 = 729 =2 x 36/64 = 784 =2 x 38/81 = 841

FROM 312 TO 392: � 312 = � 322 = � 332 = � 342 = � 352= � 362 = � 372 = � 382= � 392 = 3 x (31+1)/12 = 3 x(32)/1 = 96/1=961 3 x (32+2)/22= 3 x 34/4 = 102/4=1024 3 x (33+3)/32=3 x 36/9 = 108/9=1089 3 x (34+4)/42=3 x 38/16 = 114/16=1156 3 x (35+5)/52=1225 3 x (36+6)/62=1296 3 x (37+7)/72=1369 3 x (38+8)/82=1444 3 x (39+9)/92=1521

Square Root

Square Root of Rational Number

The End Thank yo u