Square root and cube root is a memory chapter more than a method chapter. If you know your squares till 30 and your cubes till 20, most questions in the paper are already solved before you pick up the pen. The long division method is worth learning, but honestly you will rarely need it if the tables are in your head. The real gift in this chapter is the unit digit method for cube roots. Give me any perfect cube of up to six digits and I will tell you its root in about four seconds, and so will you once you read the steps below.
Square Root and Cube Root Formulas
Squares you must know by heart, 1 to 30
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
121, 144, 169, 196, 225, 256, 289, 324, 361, 400
441, 484, 529, 576, 625, 676, 729, 784, 841, 900
Cubes you must know by heart, 1 to 20
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000
Basic rules
v(a × b) = va × vb
v(a ÷ b) = va ÷ vb
(va)² = a
v(a²) = a
³v(a × b) = ³va × ³vb
³v(a ÷ b) = ³va ÷ ³vb
(³va)³ = a
Remember va + vb is never equal to v(a + b). This is the most common mistake in the chapter.
Square root by prime factorisation
Break the number into prime factors, pair them up, and take one factor from each pair.
For a cube root, group them in threes and take one factor from each group.
If a factor is left without a pair, the number is not a perfect square, and that leftover factor is what you multiply or divide by to make it one.
Decimals, count the places
For a square root, the decimal places get halved. v0.0009 = 0.03, because 0.0009 has four decimal places and the root has two.
For a cube root, the decimal places get divided by three. ³v0.008 = 0.2, because 0.008 has three decimal places and the root has one.
A perfect square decimal must have an even number of decimal places. A perfect cube decimal must have a multiple of three.
Approximating a root that is not exact
vN is roughly a + (N – a²) ÷ (2a), where a is the nearest whole number whose square is close to N.
Example, v51 is roughly 7 + (51 – 49) ÷ 14 = 7.14
For approximation questions, always find the two whole numbers the root sits between first, then narrow down.
Nested and repeating roots
v(a × v(a × v(a × …))) continuing forever = a
v(a + v(a + v(a + …))) continuing forever = the positive solution of x² – x – a = 0
v(a – v(a – v(a – …))) continuing forever = the positive solution of x² + x – a = 0
For the common case v(6 + v(6 + v(6 + …))), the answer is 3.
60 Square Root and Cube Root Aptitude Questions and Answers (Solved MCQs)
Question 1. What is the square root of 1,04,976?
a) 314
b) 324
c) 334
d) 344
Answer:
b) 324 — Since 324² = (300 + 24)² = 90,000 + 14,400 + 576 = 1,04,976, we get √1,04,976 = 324.
Question 2. What is the value of √(2401/3481)?
a) 47/57
b) 49/57
c) 47/59
d) 49/59
Answer:
d) 49/59 — Since 2,401 = 49² and 3,481 = 59², √(2401/3481) = 49/59.
Question 3. What is the square root of 0.000729?
a) 0.0027
b) 0.027
c) 0.27
d) 2.7
Answer:
b) 0.027 — We have 0.027² = 0.000729. Therefore, √0.000729 = 0.027.
Question 4. What is the least number that must be subtracted from 5,632 to make it a perfect square?
a) 5
b) 7
c) 9
d) 11
Answer:
b) 7 — The greatest perfect square below 5,632 is 75² = 5,625. Therefore, the required number = 5,632 – 5,625 = 7.
Question 5. What is the least number that must be added to 1,750 to make it a perfect square?
a) 12
b) 14
c) 16
d) 18
Answer:
b) 14 — Since 41² = 1,681 and 42² = 1,764, the next perfect square after 1,750 is 1,764. Required number = 1,764 – 1,750 = 14.
Question 6. How many digits are there in the square root of 1,234,321?
a) 3
b) 4
c) 5
d) 6
Answer:
b) 4 — The number 1,234,321 contains seven digits. The square root of a seven-digit perfect square contains four digits. In fact, 1,111² = 1,234,321.
Question 7. What is the square root of 99,856?
a) 306
b) 312
c) 316
d) 326
Answer:
c) 316 — Using (300 + 16)², we get 300² + 2 × 300 × 16 + 16² = 90,000 + 9,600 + 256 = 99,856. Therefore, √99,856 = 316.
Question 8. What is the value of √1,764 + √2,304?
a) 86
b) 88
c) 90
d) 92
Answer:
c) 90 — Since √1,764 = 42 and √2,304 = 48, the required value = 42 + 48 = 90.
Question 9. If √x = 37, what is the value of √(9x)?
a) 101
b) 108
c) 111
d) 117
Answer:
c) 111 — Since √(9x) = √9 × √x = 3 × 37, the required value is 111.
Question 10. What is the geometric mean between 324 and 576?
a) 414
b) 424
c) 432
d) 442
Answer:
c) 432 — The geometric mean of two numbers a and b is √(ab). Therefore, √(324 × 576) = √324 × √576 = 18 × 24 = 432.
Question 11. What is the least number by which 540 must be multiplied to make the product a perfect square?
a) 10
b) 12
c) 15
d) 30
Answer:
c) 15 — Prime factorisation gives 540 = 2² × 3³ × 5. To make all exponents even, it must be multiplied by 3 × 5 = 15. The product is 8,100 = 90².
Question 12. What is the least number by which 52,920 must be divided so that the quotient becomes a perfect square?
a) 15
b) 21
c) 30
d) 42
Answer:
c) 30 — Prime factorisation gives 52,920 = 2³ × 3³ × 5 × 7². Dividing by 2 × 3 × 5 = 30 leaves 2² × 3² × 7² = 1,764 = 42².
Question 13. What is the value of √[(6.25 × 0.0144) ÷ 0.0009]?
a) 5
b) 8
c) 10
d) 12
Answer:
c) 10 — First, 0.0144 ÷ 0.0009 = 16. Therefore, the expression becomes √(6.25 × 16) = √100 = 10.
Question 14. If √(x + 144) = √x + 6, what is the value of x?
a) 64
b) 81
c) 100
d) 121
Answer:
b) 81 — Squaring both sides gives x + 144 = x + 12√x + 36. Therefore, 12√x = 108, so √x = 9 and x = 81.
Question 15. What is the square root of 47,089?
a) 207
b) 213
c) 217
d) 223
Answer:
c) 217 — Since 217² = (200 + 17)² = 40,000 + 6,800 + 289 = 47,089, the square root of 47,089 is 217.
Question 16. What is the cube root of 2,50,047?
a) 57
b) 61
c) 63
d) 67
Answer:
c) 63 — Since 63³ = 63 × 63 × 63 = 3,969 × 63 = 2,50,047, we get ∛2,50,047 = 63.
Question 17. What is the value of ∛(2744/4913)?
a) 12/17
b) 14/17
c) 14/19
d) 16/19
Answer:
b) 14/17 — Since 2,744 = 14³ and 4,913 = 17³, ∛(2744/4913) = 14/17.
Question 18. What is the cube root of 0.000216?
a) 0.006
b) 0.06
c) 0.6
d) 6
Answer:
b) 0.06 — Since 0.06³ = 0.000216, the cube root of 0.000216 is 0.06.
Question 19. What is the least number that must be subtracted from 6,890 to make it a perfect cube?
a) 21
b) 29
c) 31
d) 39
Answer:
c) 31 — The greatest perfect cube below 6,890 is 19³ = 6,859. Therefore, the required number = 6,890 – 6,859 = 31.
Question 20. What is the least number that must be added to 11,950 to make it a perfect cube?
a) 197
b) 207
c) 217
d) 227
Answer:
c) 217 — Since 22³ = 10,648 and 23³ = 12,167, the next perfect cube after 11,950 is 12,167. Required number = 12,167 – 11,950 = 217.
Question 21. What is the cube root of 3,89,017?
a) 67
b) 71
c) 73
d) 77
Answer:
c) 73 — Since 73³ = 73 × 73 × 73 = 5,329 × 73 = 3,89,017, the cube root is 73.
Question 22. What is the least number by which 675 must be multiplied to make the product a perfect cube?
a) 3
b) 5
c) 9
d) 15
Answer:
b) 5 — Prime factorisation gives 675 = 3³ × 5². To make the exponent of 5 a multiple of 3, the number must be multiplied by 5. The product is 3,375 = 15³.
Question 23. What is the least number by which 54,000 must be divided so that the quotient becomes a perfect cube?
a) 2
b) 3
c) 4
d) 5
Answer:
a) 2 — Prime factorisation gives 54,000 = 2⁴ × 3³ × 5³. Dividing by 2 leaves 2³ × 3³ × 5³ = 27,000 = 30³.
Question 24. What is the value of ∛(0.001728 ÷ 0.000008)?
a) 4
b) 5
c) 6
d) 8
Answer:
c) 6 — We have 0.001728 ÷ 0.000008 = 216. Therefore, ∛216 = 6.
Question 25. If ∛x = 12, what is the value of ∛(125x)?
a) 48
b) 50
c) 60
d) 72
Answer:
c) 60 — Since ∛(125x) = ∛125 × ∛x = 5 × 12, the required value is 60.
Question 26. What is the value of ∛1,75,616 + ∛1,10,592?
a) 96
b) 100
c) 104
d) 108
Answer:
c) 104 — Since 56³ = 1,75,616 and 48³ = 1,10,592, the required value = 56 + 48 = 104.
Question 27. The volume of a cube is 3,89,017 cubic centimetres. What is the length of each edge?
a) 69 cm
b) 71 cm
c) 73 cm
d) 75 cm
Answer:
c) 73 cm — The edge of a cube is the cube root of its volume. Therefore, the edge = ∛3,89,017 = 73 cm.
Question 28. If x is positive and ∛(x + 98) = ∛x + 2, what is the value of x?
a) 8
b) 27
c) 64
d) 125
Answer:
b) 27 — Let ∛x = y. Then ∛(x + 98) = y + 2. Cubing gives y³ + 98 = (y + 2)³. Thus, 6y² + 12y – 90 = 0, or y² + 2y – 15 = 0. Since x is positive, y = 3. Therefore, x = 3³ = 27.
Question 29. What is the least number by which 1,372 must be multiplied to make it a perfect cube?
a) 2
b) 4
c) 7
d) 14
Answer:
a) 2 — Prime factorisation gives 1,372 = 2² × 7³. Multiplying it by 2 makes every exponent a multiple of 3. Thus, 1,372 × 2 = 2,744 = 14³.
Question 30. What is the cube root of 1,28,12,904?
a) 224
b) 228
c) 234
d) 244
Answer:
c) 234 — Since 234³ = 234 × 234 × 234 = 54,756 × 234 = 1,28,12,904, the cube root of 1,28,12,904 is 234.
Question 31. What is the value of √(1764/2401) × ∛(3375/4096)?
a) 15/28
b) 45/56
c) 21/32
d) 35/48
Answer:
b) 45/56 — We have √(1764/2401) = 42/49 = 6/7 and ∛(3375/4096) = 15/16. Therefore, the required value = 6/7 × 15/16 = 90/112 = 45/56.
Question 32. What is the value of √(∛2,62,144)?
a) 4
b) 6
c) 8
d) 16
Answer:
c) 8 — Since 2,62,144 = 8⁶, its cube root is 8² = 64. Therefore, √(∛2,62,144) = √64 = 8.
Question 33. If √x = 3∛x and x is positive, what is the value of x?
a) 243
b) 512
c) 729
d) 1,024
Answer:
c) 729 — We have x^(1/2) = 3x^(1/3). Dividing by x^(1/3) gives x^(1/6) = 3. Raising both sides to the sixth power gives x = 3⁶ = 729.
Question 34. What is the least number by which 432 must be multiplied so that the product becomes both a perfect square and a perfect cube?
a) 54
b) 72
c) 108
d) 216
Answer:
c) 108 — A number that is both a perfect square and a perfect cube must be a perfect sixth power. Since 432 = 2⁴ × 3³, it must be multiplied by 2² × 3³ = 108. The product is 46,656 = 6⁶.
Question 35. What is the least number by which 5,832 must be divided so that the quotient becomes both a perfect square and a perfect cube?
a) 4
b) 6
c) 8
d) 12
Answer:
c) 8 — Prime factorisation gives 5,832 = 2³ × 3⁶. Dividing by 2³ = 8 leaves 3⁶ = 729, which is both a perfect square and a perfect cube.
Question 36. If √(x + 52) – √x = 2 and x is positive, what is the value of x?
a) 100
b) 121
c) 144
d) 169
Answer:
c) 144 — Rearranging gives √(x + 52) = √x + 2. Squaring both sides, x + 52 = x + 4√x + 4. Therefore, 4√x = 48, so √x = 12 and x = 144.
Question 37. If ∛(x + 218) – ∛x = 2 and x is positive, what is the value of x?
a) 64
b) 125
c) 216
d) 343
Answer:
b) 125 — Let ∛x = y. Then ∛(x + 218) = y + 2. For y = 5, x = 125 and x + 218 = 343 = 7³. Thus, ∛343 – ∛125 = 7 – 5 = 2.
Question 38. What is the value of √1,764 ÷ ∛13,824?
a) 5/4
b) 3/2
c) 7/4
d) 9/4
Answer:
c) 7/4 — Since √1,764 = 42 and ∛13,824 = 24, the required value = 42/24 = 7/4.
Question 39. The length of the space diagonal of a cube is 12√3 cm. What is its volume?
a) 1,296 cm³
b) 1,728 cm³
c) 2,197 cm³
d) 2,744 cm³
Answer:
b) 1,728 cm³ — The space diagonal of a cube with edge a is a√3. Therefore, a√3 = 12√3 gives a = 12 cm. Hence, volume = 12³ = 1,728 cm³.
Question 40. The diagonal of a square is 34√2 cm. What is its area?
a) 1,024 cm²
b) 1,089 cm²
c) 1,156 cm²
d) 1,225 cm²
Answer:
c) 1,156 cm² — The diagonal of a square with side a is a√2. Therefore, a√2 = 34√2 gives a = 34 cm. Hence, area = 34² = 1,156 cm².
Question 41. What is the simplified value of √(144 + √20,736)?
a) 6√2
b) 8√2
c) 10√2
d) 12√2
Answer:
d) 12√2 — Since √20,736 = 144, the expression becomes √(144 + 144) = √288 = √(144 × 2) = 12√2.
Question 42. What is the value of √6,561 – ∛91,125?
a) 32
b) 34
c) 36
d) 38
Answer:
c) 36 — Since √6,561 = 81 and ∛91,125 = 45, the required value = 81 – 45 = 36.
Question 43. If x is a positive number such that √x = 64, what is the value of √x + ∛x?
a) 72
b) 76
c) 80
d) 84
Answer:
c) 80 — Since √x = 64, x = 64² = 4,096. Also, ∛4,096 = 16. Therefore, √x + ∛x = 64 + 16 = 80.
Question 44. The total surface area of a cube is 3,456 cm². What is its volume?
a) 10,648 cm³
b) 12,167 cm³
c) 13,824 cm³
d) 15,625 cm³
Answer:
c) 13,824 cm³ — The total surface area of a cube is 6a². Thus, 6a² = 3,456 gives a² = 576 and a = 24 cm. Therefore, volume = 24³ = 13,824 cm³.
Question 45. What is the value of √83,521 – ∛2,74,625?
a) 214
b) 220
c) 224
d) 234
Answer:
c) 224 — Since √83,521 = 289 and ∛2,74,625 = 65, the required value = 289 – 65 = 224.
Question 46. What is the simplified value of √(625 + √3,90,625)?
a) 15√2
b) 20√2
c) 25√2
d) 30√2
Answer:
c) 25√2 — Since √3,90,625 = 625, the expression becomes √(625 + 625) = √1,250 = √(625 × 2) = 25√2.
Question 47. If x is a positive number that is both a perfect square and a perfect cube, and √x + ∛x = 36, what is the value of x?
a) 216
b) 512
c) 729
d) 1,296
Answer:
c) 729 — Since x is both a perfect square and a perfect cube, let x = a⁶. Then √x = a³ and ∛x = a². Thus, a³ + a² = 36. For a = 3, 27 + 9 = 36. Therefore, x = 3⁶ = 729.
Question 48. What is the least number that must be added to 99,999 to make it a perfect square?
a) 389
b) 421
c) 490
d) 512
Answer:
c) 490 — Since 316² = 99,856 and 317² = 1,00,489, the next perfect square after 99,999 is 1,00,489. Therefore, the required number = 1,00,489 – 99,999 = 490.
Question 49. What is the least number that must be subtracted from 2,50,000 to make it a perfect cube?
a) 10,648
b) 11,672
c) 12,167
d) 13,824
Answer:
b) 11,672 — Since 62³ = 2,38,328 and 63³ = 2,50,047, the greatest perfect cube below 2,50,000 is 2,38,328. Therefore, the required number = 2,50,000 – 2,38,328 = 11,672.
Question 50. How many positive integers not exceeding 10,00,000 are both perfect squares and perfect cubes?
a) 8
b) 9
c) 10
d) 12
Answer:
c) 10 — A number that is both a perfect square and a perfect cube must be a perfect sixth power. The numbers are 1⁶, 2⁶, 3⁶, …, 10⁶. Since 10⁶ = 10,00,000, there are 10 such positive integers.
Question 51. If √a = ∛b = 12, what is the value of √(ab)?
a) 144√3
b) 216√3
c) 288√3
d) 432√3
Answer:
c) 288√3 — From √a = 12, we get a = 144. From ∛b = 12, we get b = 1,728. Therefore, √(ab) = √144 × √1,728 = 12 × 24√3 = 288√3.
Question 52. If √x × ∛x = 243 and x is positive, what is the value of x?
a) 243
b) 512
c) 729
d) 2,187
Answer:
c) 729 — We have x^(1/2) × x^(1/3) = x^(5/6) = 243 = 3⁵. Therefore, x = (3⁵)^(6/5) = 3⁶ = 729.
Question 53. What is the value of ∛(√46,656)?
a) 4
b) 6
c) 8
d) 12
Answer:
b) 6 — Since 46,656 = 6⁶, √46,656 = 6³ = 216. Therefore, ∛216 = 6.
Question 54. A solid cube with an edge of 12 cm is cut into eight identical smaller cubes. By how much does the total surface area increase?
a) 432 cm²
b) 576 cm²
c) 864 cm²
d) 1,728 cm²
Answer:
c) 864 cm² — The original surface area is 6 × 12² = 864 cm². Each smaller cube has an edge of 6 cm, so their total surface area is 8 × 6 × 6² = 1,728 cm². The increase is 1,728 – 864 = 864 cm².
Question 55. What is the greatest five-digit number that is both a perfect square and a perfect cube?
a) 15,625
b) 32,768
c) 46,656
d) 59,049
Answer:
c) 46,656 — A number that is both a perfect square and a perfect cube is a perfect sixth power. We have 6⁶ = 46,656, while 7⁶ = 1,17,649, which is a six-digit number. Therefore, 46,656 is the greatest five-digit number satisfying both conditions.
Question 56. What is the least perfect cube that is exactly divisible by 1,764?
a) 37,044
b) 46,656
c) 74,088
d) 1,48,176
Answer:
c) 74,088 — Prime factorisation gives 1,764 = 2² × 3² × 7². To make every exponent a multiple of 3, multiply by 2 × 3 × 7 = 42. Thus, the least perfect cube = 1,764 × 42 = 74,088 = 42³.
Question 57. If √x + √(x/4) = 45 and x is positive, what is the value of x?
a) 625
b) 784
c) 900
d) 1,024
Answer:
c) 900 — Since √(x/4) = √x/2, the equation becomes √x + √x/2 = 45. Thus, 3√x/2 = 45, giving √x = 30. Therefore, x = 900.
Question 58. If ∛x + ∛(x/8) = 18 and x is positive, what is the value of x?
a) 1,000
b) 1,331
c) 1,728
d) 2,197
Answer:
c) 1,728 — Since ∛(x/8) = ∛x/2, the equation becomes ∛x + ∛x/2 = 18. Thus, 3∛x/2 = 18, giving ∛x = 12. Therefore, x = 12³ = 1,728.
Question 59. A positive integer x is both a perfect square and a perfect cube. If √x – ∛x = 48, what is the value of x?
a) 729
b) 1,296
c) 4,096
d) 15,625
Answer:
c) 4,096 — Since x is both a perfect square and a perfect cube, let x = a⁶. Then √x = a³ and ∛x = a². Therefore, a³ – a² = 48, or a²(a – 1) = 48. This is satisfied by a = 4. Hence, x = 4⁶ = 4,096.
Question 60. What is the value of (√1,04,976 + ∛2,50,047) ÷ √81?
a) 39
b) 41
c) 43
d) 45
Answer:
c) 43 — Since √1,04,976 = 324, ∛2,50,047 = 63 and √81 = 9, the required value = (324 + 63) ÷ 9 = 387 ÷ 9 = 43.
