Grade 8 Square and Square Roots

Transcription

1 ID : ae-8-square-and-square-roots [1] Grade 8 Square and Square Roots For more such worksheets visit Answer t he quest ions (1) The total population of a village is a perf ect square. The number of men in the village is 18956, and the number of women in the village is If we know that the number of children in the village is also a perf ect square and is more than 9400, then what is the smallest possible number of children in the village? (2) Which of the f ollowing numbers cannot be the area of a square that has an integer value side? (3) If the square root of 2 is 1.41, then f ind the square root of 50. (4) Find the value of the f ollowing expression: (5) Which is the smallest perf ect square that can be divided by 77 and 91? (6) Which of the f ollowing numbers is a perf ect square? 882, 700, 196, 252 Choose correct answer(s) f rom given choice (7) Which is the smallest number that can be used to divide to give a perf ect square? a. 22 b. 21 c. 25 d. 11 (8) There are 3015 soldiers in a platoon. The commander of the platoon wants to impress the visiting general by arranging his soldiers in such a manner that they are arranged in n rows, and each row has n soldiers. He f inds that he cannot do this with the number of soldiers he has. What is the minimum number of soldiers he needs additionally to do this kind of a f ormation? a. 13 b. 5 c. 7 d. 10 (9) What is the sum of the f irst 187 odd numbers? a b c d (10) What is the square root of 1225? a. 37 b. 64 c. 35 d. 33

2 (11) What is the value of? ID : ae-8-square-and-square-roots [2] a b c (12) What is the square root of ? d a. ±1.46 b. ±0.06 c. ±0.006 d. ±0.36 (13) Which of the f ollowing statements is always true: a. All numbers that end with '00' will be perf ect b. The dif f erence of the squares of two squares consecutive natural numbers is equal to their sum c. A number that leaves a remainder of 2 when divided by 4 is a perf ect square d. A number that has 5 in the units place can never be a perf ect square Fill in the blanks (14) Find largest possible value of (15) The square root of 76 correct to two places of decimal is 2016 Edugain ( All Rights Reserved Many more such worksheets can be generated at

3 Answers ID : ae-8-square-and-square-roots [3] (1) 9409 Total population of the village is a perf ect square. Number of men in the village Number of women in the village It is given that the number of children in the village is also a perf ect square, and is more than Let us f irst f ind the values of perf ect squares more than Square root of and the perf ect squares more than 9400 will be the squares of all integers more than , ie 97 2, 98 2, 99 2, etc. The population of children will be the smallest value out of 97 2, 98 2, 99 2, etc, f or which the total population of the village is a perf ect square. Step 5 Let us see if the population of the children can be 97 2 ( 9409). In this case, the total population of the village Since the square root of is 243, which is an integer, the total population of the village in this case is indeed a perf ect square. Step 6 Theref ore, the smallest possible number of children in the village is

4 (2) 2394 ID : ae-8-square-and-square-roots [4] We know the area of square side 2 Since it is given that side has an integer value, the area must be a perf ect square. Let's check if given areas are perf ect square or not Let's check if 1 st area of 441 is perf ect square or not using division method, is a perf ect square, theref ore it could be the are of given square Let's check if 2 nd area of 2394 is perf ect square or not using division method, is not a perf ect square, theref ore it cannot be the are of given square Let's check if 3 rd area of 3721 is perf ect square or not using division method, is a perf ect square, theref ore it could be the are of given square Step 5 Let's check if 4 th area of 2401 is perf ect square or not using division method, 49

5 ID : ae-8-square-and-square-roots [5] is a perf ect square, theref ore it could be the are of given square Step 6 Theref ore, 2394 cannot be the area of given square. (3) 7.05 We have been asked to f ind the square root of 50 if square root of 2 is Now (50) (25 2) ((5) 2 2) 5 (2) Theref ore, square root of 50 is 7.05 if square root of 2 is (4) ±22 We have been asked to f ind the value of This is an arithmetic sequence and the sum of the f irst n terms is given by: S n n(a 1 + a n ) 2 where, n number of terms, a 1 f irst term, a n last term S 22 ( 22(1 + 43) 2 ) ( ±22 ) Theref ore, the value of is +22 or -22

6 (5) ID : ae-8-square-and-square-roots [6] According to the question, we have to f ind the smallest perf ect square that can be divided by 77 and 91. 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. Prime f actors of 77 and 91 are To 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 Theref ore, the smallest perf ect square that can be divided by 77 and 91 is

7 (6) 196 ID : ae-8-square-and-square-roots [7] 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 Prime f actors of 882, Now create pairs, 882 (3 3) (7 7) 2 Since all f actors are not paired, this is not a perf ect square. Prime f actors of 700, Now create pairs, 700 (2 2) (5 5) 7 Since all f actors are not paired, this is not a perf ect square. Prime f actors of 196, Now create pairs, 196 (2 2) (7 7) Since all f actors are paired, this is a perf ect square. Step 5 Prime f actors of 252, Now create pairs, 252 (2 2) (3 3) 7 Since all f actors are not paired, this is not a perf ect square. Step 6 Theref ore correct answer is 196.

8 (7) a. 22 ID : ae-8-square-and-square-roots [8] 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. Prime f actors of Lets' create pairs of prime f actors, (2 2) (3 3) (17 17) 2 11 This number is not the perf ect square, since all f actors are not paired. If we divide this number by remaining prime f actors (i.e ), all prime f actors will be paired and number will be perf ect square. Step 5 Theref ore, 22 is the smallest number that can be used to divide to give the perf ect square. (8) d. 10 It is given that a platoon has 3015 soldiers.the number of rows and the soldiers in each row is the same. We have to f ind the number of more soldiers that should be there, so that when the commander arranged them, the number of rows and the number of soldiers in each row are same. That is, the number which should be added to 3015 to make it a perf ect square has to be calculated. If the square root of 3015 is calculated then the quotient and the remainder is 54, 99 respectively.it represents that the square of 54 with the sum of 99 is equal to the The next number is 55 and Hence the number to be added to 3015 to make it a perf ect square Theref ore, required number of soldiers is 10.

9 (9) d ID : ae-8-square-and-square-roots [9] 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 187 odd numbers. Sum of f irst n odd numbers n 2 (where, n 187) (187) Theref ore the sum of f irst 187 odd numbers is (10) c. 35 We have been asked to f ind the square root of Prime f actors of 1225 are, Now, make pairs of prime f actors. (5 5) (7 7) There is no prime f actor lef t which is not part of a pair. Square root is given by the multiplication of e element f rom each pair of prime f actors. i.e Step 5 Theref ore, the square root of 1225 is 35.

10 (11) d We have been asked to f ind the value of. ID : ae-8-square-and-square-roots [10] Now, Theref ore, the value of is (12) b. ±0.06 Let's write as: Let's f ind the square root of (6) 2 (100) 2 ±6 ±100 ±0.06 Thus the square root of is ±0.06. (13) b. The dif f erence of the squares of two consecutive natural numbers is equal to their sum

11 (14) 42 ID : ae-8-square-and-square-roots [11] Lets assume largest possible value of expression is S. S S (±9) + (±33) S will be larger if f irst number is positive, S (9) + (±33) Similarly, S will be larger if second number is positive, S (9) + (33) S 42 Thus, largest possible value is 42 (15) 8.72