4 ~ Square and Square Root Square A square of a number is that number raised to the power of 2. or When we multiply a number itself, the result we get is the square of the number multiplied. eg. 2x2=2?=4; 4is the square of 2. 3x3 =37=9; 9 is the square of 3 Perfect Square A number is said to be perfect square, if it is a square of any natural number. eg. 4x4=16, 5x5=25 Here, 16 and 25 are perfect squares of 4 and 5, respectively Square Root The square root of a number is that factor of the number which, when multiplied by itself will give that number. eg. 5x5=25, 8x8 =64 The numbers 25 and 64 are the squares of 5 and 8, respectively. We can also say that the numbers 5 and 8 are the square roots of 25 and 64, respectively. If 4? =16, we say that square root of 16 is 4 and we write Vi6 = 4 Methods for Finding Square Root 1. Prime Factorisation method 2. Division method Prime Factorisation Method ‘This method is suitable for finding square root of small numbers, eg. Vide 2 | 144 2) 72 2 36 2) 18 1 W144 = V2x2x2x2%3%3 2? x2? x3? =2x2%3=12 Example 1 Find the square root of 1089. (33 (b) 23 (43 (a)53 Sol.(a)_ Prime factor of 1089 =11 x11x3%3 => i089 = VET x11x3%3 Now, taking one number from each pair and multiplying them, we get V1089 =11 x3 = 33 Division Method This method is suitable for finding square root of, large numbers. eg. vi51321Steps of Division Method © Make the pair of digits, starting from unit digit, * Think of a number whose square is just equal to o less than first pair. ‘* Make this number as the divisor and the part of quotient and subtract its square from first pair. * Now bring the second pair down, adjacent to the remainder as new dividend. ‘+ For new trial divisor, double the quotient and enter a digit to be put at the unit place in the new divisor, such that the product of entire divisor with that digits less than equal to the dividend. ‘Repeat these steps till the last pair is taken up. 389 3 [ie1sar 9 68 [613 544 769 | 6921 e22t 151321 =389 Example 2 Find the square root of 18769. (@) 17 (b) 137 17 (a) 133 Solo) 497 1 [16769 1 23 | 7 69 267| 1869 1969 c 18769 =137 Example 3 \? +136. ce (a) 1936 (b) 4624 (©) 4196 (a) 4096 Sol.) J? +136 = of 320 = WP et36=3x320 = v? =5 x 40-136 = v= 200-136 => 7 =(64)? 096 Example 4 A gardener plants 4225 trees in such a way that there are as many rows as there are trees in a row. Find the number of trees in a row. @)6 (byes ‘Sol. (a) Number of rows (978 (am05 (75 So, number of rows = 65. — Helping Tips ‘+ If number is given in form of fraction, then find the square root of numerator and denominator 3 ronson 09 [2 = 38 8 ‘+ If number is given in decimal form, then convert it in traction form and then find the square root. yey eg, vou = [2 = 9 Vi00@ Practice Exercise — 1 if V529 =x, then find the value of x fg 97 ras ibis 6. Simplify the expression 167 x36 (27 (a3 ” 9 (ayt6 (by (245 (a)255 2. Square root of 112. is 2 a iD io 7. What is the greatest number of 4 digits, (35 Oleg Which is perfect square? 8 1 (a) 9018 (b) 9999 5s (37 (c) 9801 (a) 9899 3. If VB +V64 =x, then find the value of x. iS we +2401 =x. Fingneame ots a) 17 (b) 18. a ‘3e (ays (c) 16 (a) 14 4. 11900 + V0.09 - 0.000009 = x, then find _- Find the value of expression the value of x. 156124 i544 225. (a) 30.297 (b) 3.03 (a) 625 (by 125 (e)25—(a)225 (©) 30.003 (a) 30.300 | iC . 10. Find the least number from which, S. If 48./x +32Vx = 320, then find the value multiplication of 2352 gives a perfect of x. square number. (a9 (by (p25 (a) 36 (a2 (b)3 (HA Answers 1l@[2lo [slats lols lolelol7[ols[alo lm [wl Hints & Solutions 23 2/529 4 43129 129 x, 4529 =23 2, {n2 - pares Vos Var 1 10 aos 81+ V6a YDGMBG + |DDDDD 342 = 94+ 8=17 (900 + /0.09 - 0.000009 [s ,[-9— Yi00 * ¥1000000 +o 5 10 1000 =30 + 0.3 0.003 = 30.297 48x + 32Vx = 320 Vx(48+ 32) = 320 B80vx = 320 Xx =4, square of both sides = 4900+ vuus x=4?=163) 4 5) 6) “y wr Renraeso7 = issue KS fio 27 56 = EEN 5g = 5612+ Vie 144 144 Yib6I2+B = i505 = Pl 6 =125 Via fo wo 49 2 | 2050 26-2245 2 [te Be" 2 2 urs . Greatest number of 4 digits = 9999 21% We find that, 9999 = (99)* +198 31 aa7 So, (99? = 9999-198 = 9801 ae 2401 = VTXTXTHT = 7x7 al? x= 7649 = VS 1 aise ses FS 2352 = IXBxIKBX3xTRT «+ Required number = 3 = is12+ Jisa15 176. then find the value of x. 7) simpity [S25 511 Ya ) SPI 64 12 @4 6 8s wre @2 oe ©3 «9 Find the square root of 7921 8) simpy 1225, 2B 324 @7 (8 (89) 8 16 15 “18 [sea (a) 15 (0) 16 18 (12 Simply the expression | ‘aie a 9) Find the value of @3 9 =e we 770000 It Vx-+ 484 = 40, then find the value of xis @aZ mS © @t (325 ()415 (0) 324 (A) 525 oD v7 O 0 Tos 0.280 1% [22 «4 men seat 10) What is the value of | O=S-2 11 (1575 (0) 1521 (12968. (2116 oP oo we What is the least number of 4 cigits, which is perfect square? Answers. (@) 1089 (0) 1024 ? 3 s@ 5 {a1 yom 10 20 @ « @ 6H 79 80 9 wa