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ID : ru-8-Square-and-Square-Roots [1] Grade 8 Square and Square Roots For more such worksheets visit www.edugain.com Answer t he quest ions (1) If you subtract a number x f rom 41 times that number, and then take the square of the dif f erence, what will the result be? (2) What is the value of (3) What is the smallest number that 72 must be multiplied with to get a perf ect square? . Choose correct answer(s) f rom given choice (4) Which is the smallest number that can be multiplied to 20592 to give a perf ect square? (5) a. 141 b. 39 c. 33 d. 143 Which of the f ollowing numbers is a perf ect square? a. 1575 b. 1225 c. 980 d. 700 (6) What is the square root of 0.0225? a. ±0.015 b. ±0.45 c. ±0.15 d. ±3.15 (7) For any given number x, which of the f ollowing can be pythagorean triplets? (8) a. 2m, m2-1,m2+1 b. 2m+1, m2-1,m2+1 c. 2m, m2,m2+1 d. 2m, m2-1,m2 What is the expansion of (a - b)2? a. a2 + 2ab + b 2 b. a2 + 2ab + b c. a2 - 2ab + b 2 d. a2 - ab + b 2 Fill in t he blanks (9) (10) T he smallest perf ect square that can be divided by 77 and 14 is T he sum of the f irst 43 odd numbers is (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [2] (11) T he smallest f raction that must be added to the square root of 8 8 so that the resulting 49 number is a positive whole number is (12) T here are two squares, and the ratio of their perimeters is 7:10 If the area of the f irst square is 1225 m2, then the area of the second is m2. (13) T here are two numbers such that the dif f erence of their squares is 155 and sum of the numbers is 31, their dif f erence = (14) (15) . If the square root of 2 is 1.41, then the square root of 200 is T he square root of 32 correct to one place of decimal is © 2016 Edugain (www.edugain.com). All Rights Reserved (C) 2016 Edugain (www.Edugain.com) Many more such worksheets can be generated at www.edugain.com Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [3] Answers (1) 1600x2 Step 1 According to the question, f irst number is x and second number is 41 x. Step 2 Dif f erence of two numbers = (41 x - x) Square of the dif f erence = (41 x - x)2 = (40 x)2 = 1600 x2 Step 3 T heref ore the result will be 1600x2. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [4] (2) ±11 7 Step 1 Let's f irst f ind prime f actors of 242. 242 = 11 × 11 × 2 Step 2 Similarly f ind prime f actors of 98. 98 = 7 × 7 × 2 Step 3 Now, = √(11 × 11 × 2) √(7 × 7 × 2) = √(11 × 11 × 2) √(7 × 7 × 2) = √(11)2 √(7)2 = ±11 7 Step 4 T heref ore, the value of is ±11 . 7 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [5] (3) 2 Step 1 We have been asked to f ind the smallest number that 72 must be multiplied with to get a perf ect square. Step 2 Perf ect square is a number made by squaring a whole number.Steps to f ind the perf ect square are as f ollows. i. Resolve the given number into prime f actors. ii. Make pairs of prime f actors such that both the f actors in each pair are equal. iii. If there is nothing lef t with the pair of prime numbers so the number is perf ect square otherwise not. Step 3 Prime f actor of 72 are 2 × 2 × 2 × 3 × 3 Step 4 Lets identif y pairs of f actors, (2 × 2) × (3 × 3) × 2 Step 5 T his number is not the perf ect square.T here is a prime f actor (2) lef t with the pair of prime f actors.T o obtain the perf ect square we have to multiply the given number with the 2. Step 6 72 × 2 = 144 [Prime f actor of 144 are 2 × 2 × 2 × 2 × 3 × 3, theref ore it is the perf ect square.] Step 7 T heref ore, 2 is the smallest number that 72 must be multiplied with to get the perf ect square. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [6] (4) d. 143 Step 1 Perf ect square is a number made by squaring a whole number. Steps to check if a number is perf ect square are as f ollows. i. Resolve the given number into prime f actors. ii. Make pairs of prime f actors such that both the f actors in each pair are equal. iii. If there is nothing lef t without the pair of prime numbers so the number is perf ect square otherwise not. Step 2 Prime f actors of 20592 = 2 × 2 × 2 × 2 × 3 × 3 × 11 × 13 Step 3 Lets' create pairs of prime f actors, 20592 = (2 × 2) × (2 × 2) × (3 × 3) × 11 × 13 Step 4 T his number is not the perf ect square, since all f actors are not paired. If we multiply this number with remaining prime f actors (i.e. 11 × 13 = 143), all prime f actors will be paired and number will be perf ect square. Step 5 T heref ore, 143 is the smallest number that can be multiplied to 20592 to give the perf ect square. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [7] (5) b. 1225 Step 1 Perf ect square is a number made by squaring a whole number. Steps to f ind the perf ect square are as f ollows. i. Resolve the given number into prime f actors. ii. Make pairs of prime f actors such that both the f actors in each pair are equal. iii. If there is nothing lef t without the pair of prime numbers so the number is perf ect square otherwise not. Let's check all given numbers Step 2 Prime f actors of 1575, 1575 = 5 × 5 × 3 × 3 × 7 Now create pairs, 1575 = (5 × 5) × (3 × 3) × 7 Since all f actors are not paired, this is not a perf ect square. Step 3 Prime f actors of 1225, 1225 = 5 × 5 × 7 × 7 Now create pairs, 1225 = (5 × 5) × (7 × 7) Since all f actors are paired, this is a perf ect square. Step 4 Prime f actors of 980, 980 = 2 × 2 × 7 × 7 × 5 Now create pairs, 980 = (2 × 2) × (7 × 7) × 5 Since all f actors are not paired, this is not a perf ect square. Step 5 Prime f actors of 700, 700 = 5 × 5 × 2 × 2 × 7 Now create pairs, 700 = (5 × 5) × (2 × 2) × 7 Since all f actors are not paired, this is not a perf ect square. Step 6 T heref ore correct answer is 1225. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [8] (6) c. ±0.15 Step 1 225 Let's write 0.0225 as: 10000 Step 2 Let's f ind the square root of 225 10000 = √ 225 √ 10000 = √ 15 × 15 √ 100 × 100 = √(15)2 √(100)2 = ±15 ±100 = ±0.15 Step 3 T hus the square root of 0.0225 is ±0.15. (7) a. 2m, m2-1,m2+1 (8) c. a2 - 2ab + b 2 Step 1 If you look at the question caref ully, you will notice that you have to f ind out the expansion of (a - b)2 . Step 2 Now, (a - b)2 = (a - b)(a - b) = (a - b)a - (a - b)b = a2 -ab - ab - (-b)b = a2 - 2ab + (b)b = a2 - 2ab + b 2 Step 3 T heref ore the expansion of (a - b)2 is a2 - 2ab + b 2 . (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [9] (9) 23716 Step 1 According to the question, we have to f ind the smallest perf ect square that can be divided by 77 and 14. Step 2 Perf ect square is a number made by squaring a whole number.Steps to f ind the perf ect square are as f ollows. i. Resolve the given number into prime f actors. ii. Make pairs of prime f actors such that both the f actors in each pair are equal. iii. If there is nothing lef t with the pair of prime f actors so the number is perf ect square otherwise not. Step 3 Prime f actors of 77 and 14 are 7 × 11 × 2. T o make perf ect square the prime f actors must f ollow the condition of perf ect square. Prime f actors of smallest perf ect square = 7 × 11 × 2 × 7 × 11 × 2 = 11 × 11 × 7 × 7 × 2 × 2 = 23716 Step 4 T heref ore, the smallest perf ect square that can be divided by 77 and 14 is 23716. (10) 1849 Step 1 If you look at the question caref ully, you will notice that you have to f ind out the value of the sum of the f irst 43 odd numbers. Step 2 Sum of f irst n odd numbers = n 2 (where, n = 43) = (43)2 = 1849 Step 3 T heref ore the sum of f irst 43 odd numbers is 1849. (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [10] (11) 1 7 Step 1 Let's f irst convert mixed f raction to improper f raction 8 8 = 49 400 49 Step 2 Now, = √400 √49 ⇒ = ±20 ±7 ⇒ = 20 .......... [ Use only positive number since question is about positive whole 7 number] ⇒ =2+ 6 7 Step 3 Now f or above number to be a whole number, we just need to f ind a f raction that must me added to 6/7, such that sum if 1. T heref ore required f raction to be added, =1- 6 7 = 1 7 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [11] (12) 2500 Step 1 Let us assume the side of f irst square is a. Let us assume the side of second square is b. Perimeter of f irst square = 4a Perimeter of second square = 4b Ratio of the perimeter = 7:10 Step 2 Area of the f irst square, a2 = 1225 ⇒ a2 = 352 ⇒ a = 35 Step 3 Now the ratio of perimeter of the squares, 4a 4b ⇒ 7 = 10 35 = b 7 10 ⇒ 35 × 10 = 7 × b ⇒ 350 =b 7 ⇒ b = 50 m Step 4 Area of the second square, b2 = (50)2 = 50 × 50 = 2500 m2 Step 5 T heref ore, the area of the second square is 2500.vm2 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited ID : ru-8-Square-and-Square-Roots [12] (13) 5 Step 1 Let us assume x and y are the two numbers. Step 2 It is given that, the dif f erence of their squares is 155, that is, x2 - y2 = 155 ----(1) Step 3 It is also given that sum of the numbers is 31, that is, x + y = 31 ----(2) Step 4 We know that, x2 - y2 = (x + y)(x - y) Step 5 On substituting values f rom Eq. (1) and (2), 155 = 31 × (x - y) x - y = 155/31 x- y=5 Step 6 T heref ore, their dif f erence is 5. (14) 14.1 Step 1 We have been asked to f ind the square root of 200 if square root of 2 is 1.41. Step 2 Now √(200) = √(100 × 2) =√((10)2 × 2) = 10 × √(2) = 10 × 1.41 = 14.1 Step 3 T heref ore, square root of 200 is 14.1 if square root of 2 is 1.41. (15) 5.7 (C) 2016 Edugain (www.Edugain.com) Personal use only, commercial use is strictly prohibited