5. Square Roots and Cube Roots

Solved Examples

1. Evaluate √6084 by factorization method.2. Find the square root of 1471369.3. Evaluate : √248+√51+√169 .

4. If a * b * c = √(a+2 )(b+3)c+1

, then find the value of 6 * 15 * 3.

5. Find the value of √1 916 .

6. What is the square root of 0.0009?7. Evaluate √175.29768. What will come in place of question mark in each of the following questions?

i. √ 32.4? =2 ii. √86.49 + √5+(? )2=12.3.

9. Find the value of √ 0.2890.00121

.

10. If √1+ x144

= 1312 , then find the value of x.

Square Root: If x2 = y, we say that the square root of y is x and we write, √ y =x.

Thus, √4 =2, √9 =3, √196 =14.

Cube Root: The cube root of a given number x is the number whose cube is x. We denote the cube root of x by 3√ x .

Thus, 3√8 = 3√2×2×2 =2, 3√343 = 3√7×7×7 =7 etc.

Note:

1. √ xy = √ x × √ y 2. √ xy = √ x√ y = √ x√ y × √ y√ y = √x y√ y

Important Facts and Formulae

11. Find the value of √3 up to three places of decimal.

12. If √3 =1.732, find the value of √192 – 12 √48−√75 correct to 3 places of decimal.

13. Evaluate √ 9.5× .0085×18.9.0017×1.9×0.021

14. Simplify : √ [ (12.1 )2−(8.1 )2 ]÷ [ (0.25 )2+(0.25 ) (19.95 )] .15. If x=1 + √2 and y =1 – √2 , find the value of (x2 + y2).16. Evaluate √0.9up to 3 places of decimal.

17. If √15 = 3.88, find the value of √ 53 .

18. Find the least square number which is exactly divisible by 10,12, 15 and 18.19. Find the greatest number of five digits which is a perfect square.20. Find the smallest number that must be added to 1780 to make it a perfect square.

21. If √2 - 1.4142, find the value of √22+√2 .

22. If x = ( √5+√3√5−√3 ) and y = (√5−√3

√5+√3 ), find the value of (x2 + y2 ).

23. Find the cube root of 2744.24. By what least number 4320 be multiplied to obtain a number which is a perfect cube?

Answers:

1 78 13 1502  1213 14  43  16 15  64  3 16  0.9485  1 ¼ 17  1.2936  0.03 18  9007  13.24 19  998568 (i)8.1 (ii) 2 20  69

9 17011 21  0.4142

10  25 22  6211  1.732 23  1412  1.732 24  50

Objective Type Questions

1. √53824 =?a. 202 b. 232 c. 242 d. 332

2. The square root of 64009 is:a. 253 b. 347 c. 363 d. 803

3. The value of √10+√25+√108+√154+√225 is:

a. 4 b. 6 c. 8 d. 104. Evaluate: √41−√21+√19−√9 .

a. 3 b. 5 c. 6 d. 6.45. √176+√2401 is equal to:

a. 14 b. 15 c. 18 d. 24

6. (√62511 × 14√25

× 11√196 ) is equal to:

7. (√ 225729−√ 25144 )÷√ 1681 =?

8. The square root of (2722 – 1282 ) is: a. 144 b. 200 c. 240 d. 256

9. If x*y = x + y + √ xy, the value of 6 * 24 is:a. 41 b. 42 c. 43 d. 44

10. If y =5, then what is the value of 10y √ y3− y2 ? a. 50 √2 b. 100 c. 200 √5 d. 500

11. √110 14 =?

a. 10.25 b. 10.5 c. 11.5 d. 19.5

12. √ 2581−19 =?

a.23 b.

49 c.

1681 d.

2581

13. The digit in the unit’s place in the square root of 15876 is:a. 2 b. 4 c. 6 d. 8

14. How many two – digit numbers satisfy this property: The last digit (unit’s digit) of the square of the two-digit number is 8?

a. 1 b. 2 c. 3 d. None of these

15. What is the square root of 0.16?a. 0.004 b. 0.04 c. 0.4 d. 4

16. The value of √0.000441 is:a. 0.00021 b. 0.0021 c. 0.021 d. 0.21

17. √0.00004761 equals:a. 0.00069 b. 0.0069 c. 0.0609 d. 0.069

18. 1.52 × √0.0225 =?a. 0.0375 b. 0.3375 c. 3.275 d. 32.75

19. √0.01+√0.0064= ?a. 0.03 b. 0.3 c. 0.42 d. None of these

20. The value of √0.01+ √0.81 + √1.21 + √0.0009 is:a. 2.03 b. 0.3 c. 0.42 d. None of these

21. √ .0025 × √2.25 × √ .0001 =?a. .000075 b. .0075 c. .075 d. None of these

22. √1.5625 =?a. 1.05 b. 1.25 c. 1.45 d. 1.55

23. If √ .00000676 = .0026, the square root of 67,60,000 is:

a.126 b. 26 c. 260 d. 2600

24. If √18225 = 135, then the value of (√182.25 + √1.8225+ √0.018225+ √0.00018225 ) is:

a. 1.49985 b. 14.9985 c. 149.985 d. 1499.8525. Given that √13 =3.605 and √130 = 11.40, find the value of √1.3 + √1300 + √0.013

a. 36.164 b. 36.304 c. 37.164 d. 37.304

26. If 52x = √ 169289 , the value of x is:

a. 52 b. 58 c. 62 d. 68

27. For what value of * the statement ( ¿15 )( ¿

135 ) = 1 is true?

a. 15 b. 25 c. 35 d. 45

28. Which number can replace both the question marks in the question 4 12?

= ?32 .

a. 1 b. 7 c. 7 12 d. None of these

29. What should come in place of both the question marks in equation ?

√128 = √162?

. a. 12 b. 14 c. 144 d. 196

30. If 0.13 ÷ p2 =13, then p equals :a. 0.01 b. 0.1 c. 10 d. 100

31. What number should be divided by √0.25 to give the result as 25?a. 12.5 b. 25 c. 50 d. 125

32. If √3n = 729, then the value of n is:a. 6 b. 8 c. 10 d. 12

33. If √18×14×x =84, then x equals:a. 22 b. 24 c. 28 d. 32

34. 28√? + 1426 = 34 of 2872

a. 576 b. 676 c. 1296 d. 1444

35. √ ?169

= 5439

a. 108 b. 324 c. 2916 d. 480036. If √ x ÷ √441 =0.02, then the value of x is:

a. 0.1764 b. 1.764 c. 1.64 d. 2.64

37. √ .0196? = 0.2

a. 0.49 b. 0.7 c. 4.9 d. None of these38. √0.0169×? =1.3

a. 10 b. 100 c. 1000 d. None of these39. If √1369+ √ .0615+x = 37.25, then x is equal to:

a. 10-1 b. 10-2 c. 10-3 d. None of these40. If √ ( x−1 )( y+2) =7, x and y being positive whole numbers, then the values of x and y

respectively are:a. 8, 5 b. 15, 12 c. 22, 19 d. None of these

41. If √ .04× .4×a = .004 × .4 × √b , then ab is:

a. 16 ×10-3 b. 16 ×10-4 c. 16 ×10-5 d. None of these42. Three – fifth of the square of a certain number is 126.15. What is the number?

a. 14.5 b. 75.69 c. 145 d. 210.25

43. √ 0.3610.00169

=?

a.1.913 b.

1913 c.

1.9130 d.

19013

44. √ 48.40.289 is equal to:

a. 1 517 b. 12

117 c. 12

1617 d. 129

717

45. If √1+ x169

= 1413 , then x is equal to:

a. 1 b. 13 c. 27 d. None of these

46. If √1+ 55729 = 1 + x27 , then the value of x is:

a. 1 b. 3 c. 5 d. 747. The value of √2 up to three places of decimal is:

a. 1.410 b. 1.412 c. 1.413 d. 1.41448. (2 √27 - √75 + √12) is equal to:

a. √3 b. 2√3 c. 3 √3 d. 4 √349. By how much does √12+√18 exceeds √3 + √2 ?

a. √2 – 4 √3 b. √3 + 2√2 c. 2 (√3 – √2 ) d. 3 (√3−√2 )50. √24+√216√96 =?

a. 2 √6 b. 2 c. 6 √2 d. 2√651. The value of √80−√122

√45−√63 is:

a.34 b. 1

13 c. 1

79 d. 1

34

52. If 3√5 +√125 = 17.88, then what will be the value of √80 + 6√5 ?a. 13.41 b. 20.46 c. 21.66 d. 22.35

53. √50 × √98 is equal to:a. 63.75 b. 65.95 c. 70 d. 70.25

54. Given √2 =1.414. The value of √8+2√32−3√128+4√50 is:a. 8.426 b. 8.484 c. 8.526 d. 8.876

55. The approximate value of 3√122√28 ÷ 2√21√98 is:

a. 1.0605 b. 1.0727 c. 1.6007 d. 1.6026

56. √ .081×.484.0064×6.25 is equal to:

a. 0.9 b. 0.99 c. 9 d. 99

57. √ 0.204×420.07×3.4 is equal to:

a.16 b. 0.06 c. 0.6 d. 6

58. √ 0.081×0.324×4.6241.5624×0.0289×72.9×64

is equal to:

a. 0.024 b. 0.24 c. 2.4 d. 24

59. √ 9.5× .085.0017× .19 equals:

a. .05 b. 5 c. 50 d. 500

60. The value of √ (0.03)2+(0.21)2+(0.065)2

(0.003)2+(0.021)2+(0.0065)2 is:

a. 0.1 b. 10 c. 102 d. 103

61. The square root of (7 + 3√5) (7 – 3√5) is:a. √5 b. 2 c. 4 d. 3 √5

62. (√3− 1√3

)2

simplifies to:

a.34 b.

4√3 c.

43 d. None of these

63. (√2+ 1√2 )2 is equal to:

a. 2 12 b. 3 12 c. 4

12 d. 5

12

64. If a = 0.1039, then the value of √4 a2−4a+1 + 3a is:a. 0.1039 b. 0.2078 c. 1.1039 d. 2.1039

65. The square root of (0.75)3

1−0.75 + [0.75 + (0.75)2 + 1] is:

a. 1 b. 2 c. 3 d. 466. If 3a = 4b = 6c and a + b + c = 27 √29, then √a2+b2+c2 is:

a. 3 √29 b. 81 c. 87 d. None of these67. The square root of 0.4 is:

a. 0.6 b. 0.7 c. 0. 8 d. 0. 968. Which one of the following numbers has rational square root?

a. 0.4 b. 0.09 c. 0.9 d. 0.02569. The value of √0.4 is:

a. 0.02 b. 0.2 c. 0.51 d. 0.6370. The value of √0.121 is:

a. 0.011 b. 0.11 c. 0.347 d. 1.171. The value of √0.064 is:

a. 0.008 b. 0.08 c. 0.252 d. 0.8

72. The value of √ 0.160.4 is:

a. 0.02 b. 0.2 c. 0.63 d. None of these

73. The value of 1+√0.011−√0.1 is close to:

a. 0.6 b. 1.1 c. 1.6 d. 1.7

74. If √5 = 2.236, then the value of 1√5 is:

a. .367 b. .447 c. .745 d. None of these

75. If √24 = 4.899, the value of √ 83 is:

a. 0.544 b. 1.333 c. 1.633 d. 2.666

76. If √6 = 2.449, the value of 3√22√3

is:

a. 0.6122 b. 0.8163 c. 1.223 d. 1.2245

77. If √5 = 2.236, then the value of √52

- 10√5 + √125 is equal to:

a. 5.59 b. 7.826 c. 8.944 d.10.06278. If 2 * 3 = √13 and 3*4 = 5, then the value of 5*12 is:

a. √17 b. √29 c. 12 d. 1379. The least perfect square number divisible by 3, 4, 5, 6 and 8 is:

a. 900 b. 1200 c. 2500 d. 360080. The least perfect square, which is divisible by each of 21, 36 and 66, is:

a. 213444 b. 214344 c. 214434 d. 23144481. The least number by which 294 must be multiplied to make it a perfect square, is:

a. 2 b. 3 c. 6 d. 2482. Find the smallest number by which 5808 should be multiplied so that the product

becomes a perfect square.a. 2 b. 3 c. 7 d. 11

83. The least number by which 1470 must be divided to get a number which is a perfect square, is:

a. 5 b. 6 c. 15 d. 3084. What is the smallest number to be subtracted from 549162 in order to make it a perfect

square?a. 28 b. 36 c. 62 d. 81

85. What is the least number which should be subtracted from 0.000326 to make it a perfect square?

a. 0.000002 b. 0.000004 c. 0.02 d. 0.0486. The smallest number added to 680621 to make the sum a perfect square is:

a. 4 b . 5 c. 6 d. 887. The greatest four-digit perfect square number is:

a. 9000 b. 9801 c. 9900 d. 998188. The least number of 4 digits which is a perfect square is:

a. 1000 b. 1016 c. 1024 d. 1036

89. Given √5 = 2.2361, √3 = 1.7321, then 1

√5−√3 is equal to:

a. 1.98 b. 1.984 c. 1.9841 d. 2

90.1

√9−√8− 1

√8−√7− 1

√7−√6− 1

√6−√5+ 1

√5−√4 is equal to:

a. 0 b. 13 c. 1 d. 5

91. (2+√2+ 12+√2

+ 1√2−2 ) simplifies to:

a. 2 - √2 b. 2 c. 2 + √2 d. 2√2

92. If √2 = 1.4142, the value of 7

(3+√2) is:

a. 1.5858 b. 3.4852 c. 3.5858 d. 4.4142

93. [ 3√2√6−√3

− 4 √3√6−√2

− 6√8−√12 ] =?

a. √3−√2 b. √3+√2 c. 5 √3 d. 1

94. √7+√5√7−√5 is equal to:

a. 1 b. 2 c. 6 – √35 d. 6 + √35

95. If 5+2√37+4 √3 = a + b√3 , then

a. a = -11, b= -6 b. a= -11, b= 6 c. a=11, b= -6 d. a=6, b=11

96. If √2 =1.414, the square root of √2−1√2+1 is nearest to:

a. 0.172 b. 0.414 c. 0.586 d. 1.414

97. 3+√65√3−2√12−√32+√50 = ?

a. 3 b. 3√2 c. 6 d. None of these

98. ( 2+√32−√3

+2−√32+√3

+ √3−1√3+1 ) simplifies to:

a. 16 −√3 b. 4−√3 c. 2−√3 d. 2 +√3

99. If x = (7 – 4√3), then the value of (x+ 1x ) is:

a. 3√3 b. 8 √3 c. 14 d. 14 + 8√3

100. If x = √3+1√3−1 and y = √3−1√3+1 , then the value of (x2 + y2 ) is:

a. 10 b. 13 c. 14 d. 15

101. If a = √5+1√5−1 and b=√5−1√5+1 , the value of ( a2+ab+b2a2−ab+b2 ) is:

a.34 b.

43 c.

35 d.

53

102. A man plants 15376 apple 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:

a. 124 b. 126 c. 134 d. 144103. 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?a. 65 b. 81 c. 100 d. None of these

104. A group of students decided to collect as many paise from each member of the group as is the number of members. If the total collection amounts to Rs. 59.29, the number of members in the group is:

a. 57 b. 67 c. 77 d. 87105. The cube root of .000216 is:

a. .6 b. .06 c. .006 d. None of these

106. 3√4 12125 =?

a. 1 25 b. 1

35 c. 1

45 d. 2

25

107. 3√√ .000064 =?a. .02 b. .2 c. 2 d. None of these

108. The largest four-digit number which is a perfect cube, is:a. 8000 b. 9261 c. 9999 d. None of these

109. By what least number 675 be multiplied to obtain a number which is a perfect cube?a. 5 b. 6 c. 7 d. 8

110. What is the smallest number by which 3600 be divided to make it a perfect cube?a. 9 b. 50 c. 300 d. 450

Answers:

1 b 21 d 41 c 61 b 81 c2 a 22 b 42 a 62 c 82 b3 a 23 d 43 d 63 c 83 d4 c 24 b 44 c 64 c 84 d5 b 25 d 45 c 65 b 85 a6 a 26 d 46 a 66 c 86 a7 c 27 d 47 d 67 a 87 b8 c 28 d 48 c 68 b 88 c9 b 29 a 49 b 69 d 89 c

10 d 30 b 50 b 70 c 90 d11 b 31 a 51 b 71 c 91 b12 b 32 d 52 d 72 c 92 a13 c 33 c 53 c 73 c 93 c14 d 34 b 54 b 74 b 94 d15 c 35 b 55 a 75 c 95 c16 c 36 a 56 b 76 d 96 b17 b 37 a 57 d 77 b 97 d18 b 38 b 58 a 78 d 98 a19 b 39 c 59 c 79 d 99 c20 d 40 a 60 b 80 a 100 c

101 b 106 b102 a 107 b103 c 108 b104 c 109 a105 b 110 d