The document, authored by Dr. T.K. Jain, discusses square roots, number systems, and entrepreneurship, emphasizing the need for supportive institutions for aspiring entrepreneurs. It includes various mathematical problems and their solutions related to square roots and related concepts, alongside links to additional educational resources. Dr. Jain advocates for free education and training opportunities for all in the context of fostering entrepreneurship and social development.

1. SQUARE ROOTS, DECIMALS & NUMBER SYSTEM by : DR. T.K. JAIN AFTERSCHO ☺ OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india FOR – PGPSE / CSE PARTICIPANTS mobile : 91+9414430763

2. My words.... Here I present a few basic questions questions on square root and other related issues. I wish that more people should become entrepreneurs. An ordinary Indian entrepreneur wishes to remain an honest entrepreneur and contribute to the development of nation but we have to strengthen those institutions which truly promote entrepreneurship, not just degree granting institutions. Let us work together to promote knowledge, wisdom, social development and education. We believe in free education for all, free support for all, entrepreneurship opportunities and training for all. Let us work together for these goals. ... I alone cant do much, I need support of perosns like you .......... ...

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6. By what least number 4320 be multiplied to obtain number which is a perfect cube? In a cube, every factor must be in triplet. We have to chech whether each factor is in triplet or not in 4320. Let us find the factors of 4320 : =2*2*2*2*2*5*3*3*3 we have five 2, three 3, and one 5. we need to multiply it by one 2 and two 5. =2*5*5 = 50 answer

7. A general wishes to draw up his 36581 soldiers in the form of a solid square ,after arranging them ,he found that some of them are left over .How many are left? Let us find square root of 36581 divide it by : 1, we get 2 as remainder, now we have 265 we have to divide by 29, we get : (29*9) = 261 the remainder is : 481 we can now divide it by : 381,we get : 381*1 = 381 the remainder is :100 answer

8. A general wishes to draw up his 50000 soldiers in the form of a solid square ,after arranging them ,he found that some of them are left over .How many are left? Let us find square root of 50000 223 is square root and remainder is 271 so 271 soldiers are left out.

9. A general wishes to draw up his 100000 soldiers in the form of a solid square ,after arranging them ,he found that some of them are left over .How many are left? Let us find square root of 100000 remainder is 144, so this is the answer

10. A little child is making a cubic building out of small cubical wooden parts. Each part has one side of 1 inch. The child has 500 parts. How many parts will be left out after the building with maximum wooden part is ready? Options : a. 157 168 211 121 answer : a cube of 7 is 343 and cube of 8 is 512, so the child would use 343 wooden parts.

11. students from an institution collected as many paise from each students as there are number of students. Thus they collected Rs. 17344.89 How much did each student contribute ? Let us first convert this amount in paise : 1734489 now find square root of this amount 1317 so each student has contributed Rs. 13.17

12. A man plants 15376 apples trees in his garden and arranges them so, that there are as many rows as there are apples trees in each row .The number of rows is. Solution : find square root of 15376 124 answer

13. .The greatest four digit perfect square number is If we multiply 100, we get smallest 5 digit square number. So let us square up 99. to square 99 (100 – 1)^2, we get a^2-2ab+b^2 =10000 – 200 +1 =9801 answer

14. The smallest number added to 680621 to make the sum a perfect square is Find out square root of 680621 we get : 825, but its square is 4 more than this number, so let us add 4 in this number to make a perfect square.

15. Find square root of (.036*.49*.025)/(.16*.064*.081) We can remove all the decimal sounds, as there are 8 digits after decimal (6*7*5)/(4*8*9) =35/48 answer

16. If sqrt(x/169) = 54/39, what is X? Sqrtx = (54*13)/39 sqrtx = 18 x = 18*18 = 324 answer

17. Find square root of (63*15*21*125) Let us write these in the form of prime factors : (3*3*7*3*5*3*7*5*5*5) let us put the similar numbers together: (3*3*3*3*5*5*5*5*7*7) square root =9*25*7 = 1575 answer

18. Find cubic root of (60*25*6*24) In order to find cubic root of this, we must first put it in the form of prime factors : =(2*2*3*5*5*5*2*3*2*2*2*3) now put similar prime factors together: =(2*2*2*2*2*2*3*3*3*5*5*5) cubic root = 2*2*3*5 = 60 answer

19. Find quadric root of (36*20*45*25) Let us find the prime factors of each numbers and write the prime factors : =(2*2*3*3*2*2*5*3*3*5*5*5) now put the similar prime factors together: =(2*2*2*2*3*3*3*3*5*5*5*5) quadric root=2*3*5 = 30 answer

20. What is square root of (8+2*sqrt(15))? Options : a. sqrt2 + sqrt6 b. sqrt5 + sqrt3 c. 2 + sqrt6 d. 2*sqrt3 + 5*sqrt5 answer : try with options : answer = b . (use (a+b)^2 = a^2+b^2+2ab

21. What is square root of (7+2*sqrt(10))? Options : a. sqrt2 + sqrt5 b. sqrt4 + sqrt3 c. 1 + sqrt6 d. 2 + sqrt5 answer : try with options : answer =a . (use (a+b)^2 = a^2+b^2+2ab

22. What is square root of (3+sqrt(5))? Options : a. sqrt3 / 2 +1/ sqrt2 b. sqrt(5/2) + sqrt(1/2) c.3 + sqrt(1/5) d. 2 + sqrt5 answer : try with options : answer =b . (use (a+b)^2 = a^2+b^2+2ab

23. (9^n*3^5*27^(1/3))/(3*10000^0*81^4) = 27, what is n? (9^n*3*3*3*3*3*3)/(3^17) =3^3 9^n=3^14 or 3^2n = 3^14 n = 7

24. 2^(x-1) +2^(x+1) = 320, what is X? Take 2^(x-1) as common factor, we get : 2^(x-1) (1 + 2^2) = 320 or 5 * 2^(x-1) = 320 or 2^(x-1) = 64 =2^6 x-1 = 6 so x =7 answer or solve using options : Options : a 6, b 8, c 5, d 7 try using options : answer d

25. 2^(x+4) – 2^(x+2) = 3, what is x? Options : 0,2,-2,-1 put x = -2 and you get the answer

26. 2^(2x-1) = 1/(8^(x-3)), what is x? Options : 3,2,0,-2 put x = 2 and you get the answer

27. 3^(4x-2) = 729, what is x? Options : 1,1.5,2,2.5, try to put the value x=2, you get the answer

28. 32^1/3 = 2^x, what is x? Options : 5,3,3/5,5/3 answer :

29. 1.66666666666..... CAN BE EXPRESSED AS (FRACTION) ? Here 6 is repeated. For this type of question, write as many 9 (in denominator) as there are repeated numbers. Here we have only one repeated number, so write only one 9 as denominator 1 + 6/9 or 15/9 or 5/3 answer

30. 1.6232323232323232..... CAN BE EXPRESSED AS (FRACTION) ? Here 23 is repeated. For this type of question, write as many 9 (in denominator) as there are repeated numbers. Here we have 2 repeated numbers, so write two 9 as denominator. We have to write as many non-zero as there are non repeated digits 1 + (623-6)/990 = 1+ 617/990 =1607/990 answer

31. 3.5321321321321321321..... CAN BE EXPRESSED AS (FRACTION) ? Here 321 is repeated. For this type of question, write as many 9 (in denominator) as there are repeated numbers. Here we have 3 repeated numbers, so write three 9 as denominator. We have to write as many non-zero as there are non repeated digits 3 + (5321 – 5) / (9990) =3 + 5316/9990 = 35286/9990 answer

32. 1.34289898989..... CAN BE EXPRESSED AS (FRACTION) ? Here 89 is repeated. For this type of question, write as many 9 (in denominator) as there are repeated numbers. Here we have 2 repeated numbers, so write two 9 as denominator. We have to write as many non-zero as there are non repeated digits 1 + (34289 – 342) / (99000) =1 + 33947 / 99000 =132947 / 99000 answer

33. 1.6123412341234...... CAN BE EXPRESSED AS (FRACTION) ? Here 1234 is repeated. For this type of question, write as many 9 (in denominator) as there are repeated numbers. Here we have 4 repeated numbers, so write four 9 as denominators 1 + (61234 – 6) / 99990 =1 + (61228)/99990 = 161218 / 99990 answer

34. What is HCF of .54, .06 and 1.08? Let us multiply each of these numbers by 100, we get : 54, 6, 108 now find their HCF : it is 6, as each of these numbers is divisible by 6. Now again divide 6 by 100. = .06 answer

35. What is LCM of .54, .06 and 1.08? Let us multiply each of these numbers by 100, we get : 54, 6, 108 List down their factors : 54 = 3*3*3*2, 108=3*3*3*2*2 now find their LCM:108 Now again divide LCM by 100. = 1.08 answer

36. Express .4343434343...... as fraction? 43/99 answer

37. What will be the answer if .63636363.. is added to .37373737..? Add 63/99 + 37/99 = 100/99 answer

38. .0000000002 as scientific notation? 2 * 10^(-10)

39. What will you get from : (.5^4 - .4^4) / (.5^2 +.4^2) We know that a^2 – b^2 = (a+b) (a-b) thus we can cancel .5^2 +.4^2 we get : .25 - .16 =.09 answer

40. Put these in ascending order : 2/8, 5/-7, 18/-23 ? In ascending order, we put from small to big the smallest number here is 18/-23, here confusion is between 5/7 and 18/23, so compare them. Multiply the numerator of each number with denominator of other number, we get : 115 and 126. the bigger is 126, but the numbers are negative, so the bigger number is 115. thus -126 is smaller. so correct order will be : 18/-23, 5/-7, 2/8 answer

41. What is square root of .4? Choices : a. .2 b .02 c.002 d.none of these Answer : None of these square root of .4 is close to .6 (remember, square roots of numbers below 1 is always more than the number itself)

42. What is square root of .36? Choices : a. .6 b .06 c.006 d.none of these Answer : .6 (remember, square roots of numbers below 1 is always more than the number itself)

43. What is square root of .25? Choices : a. .5 b .05 c.005 d.none of these Answer : .5 (remember, square roots of numbers below 1 is always more than the number itself)

44. If 1/3.718 =0.2689,then find value of 1/0.0003718 ? Here decimal has been shifted 4 places to the left in the denominator. So the answer is = 2689 answer

45. If ½ =0.5,then find value of 1/0.00000002 ? Here decimal has been shifted 8 places to the left in the denominator. So the answer is =50000000 answer

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