Square root and cube root

The cube root

The cube root of a number is a special value that, when used in a multiplication three times, gives that number.

Example: 3 × 3 × 3 = 27, so the cube root of 27 is 3.

CUBES FROM 0 TO 6

0 cubed = 03 = 0 × 0 × 0 = 0

1 cubed = 13 = 1 × 1 × 1 = 1

2 cubed = 23 = 2 × 2 × 2 = 8

3 cubed = 33 = 3 × 3 × 3 = 27

4 cubed = 43 = 4 × 4 × 4 = 64

5 cubed = 53 = 5 × 5 × 5 = 125

6 cubed = 63 = 6 × 6 × 6 = 216

The square root

The square root of a number is a value that, when multiplied by itself, gives the number.
Example: 4 × 4 = 16, so a square root of 16 is 4.
Note that (−4) × (−4) = 16 too, so −4 is also a square root of 16.
The symbol is √ which always means the positive square root.
Example: √36 = 6 (because 6 x 6 = 36)

LET'S TAKE SOME EXAMPLE'S
1 . Find the square root of 484 by prime factorization method.
A. 11 B. 22 C. 33 D. 44

Answer: B

Explanation:
Resolving 484 as the product of primes, we get
484 = 2 × 2 × 11 × 11

√484 = √(2 × 2 × 11 × 11) = 2 × 11
Therefore, √484 = 22

2 . Find the cube root of 19683.
A. 25 B. 26 C. 27 D. 28

Answer: C

Explanation:
In 19683, 19 lies between 2³ and 3³, so left digit is 2 and 683 ends with 3, so right digit is 7.
thus 27 is a cube root of 19683

3 . A certain number of people agree to subscribe as many rupees each as a there are subscribers. The whole subscription is 2582449 rupees. Find the number of subscribers.
A. 1607 B. 1802 C. 2056 D. 2287

Answer: A

Explanation:
Let the total number of subscribers be x
. Then, according to the question, x² = 2582449
∴ x = √2582449
= 1607

4 . Find the square root of 324.
A. 36 B. 23 C. 18 D. 11

Answer: C

Explanation:
The square root of 324 by prime factorization, we get
324 = 2 × 2 × 3 × 3 × 3 × 3
√324 = √(2 × 2 × 3 × 3 × 3 × 3)
= 2 × 3 × 3
Therefore, √324 = 18

5 . Find the least number which when multiplied with 74088 will make it a perfect square.
A. 19 B. 22 C. 36 D. 42

Answer: D

Explanation:
74088 = 2 x 2 x 2 x 3 x 3 x 3 x 7 x 7 x 7
= (2x2) x (3x3) x (7x7) x (2x3x7)
Least number which should be multiplied to 74088 to make it a Perfect square = 2 x 3 x 7 = 42

6 . Find out the square root of 1764.
A. 41 B. 42 C. 43 D. 44

Answer: B

Explanation:
The square root of 1764 by prime factorization, we get
1764 = 2 x 2 x 3 x 3 x 7 x 7.
√1764 = √(2 x 2 x 3 x 3 x 7 x 7)
= 2 x 3 x 7
Therefore, √1764 = 42.

7 . Find the least number by which 175760 be divided to make it a perfect cube.
A. 10 B. 17 C. 23 D. 35

Answer: A

Explanation:
175760 =2³ x 2 x 5 x 13³
= 2³ x 2 x 5 x 13³
To make perfect cube, it must be divided by 2 x 5 = 10

8 . The student of class XI of a school donated 4356 Rupees in all, for Prime Minister's National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number o
A. 61 B. 66 C. 72 D. 78

Answer: B

Explanation:
Total money donated = 4356 rupees
As total students are equal to the sum donated by each student.
Total Students =√4356
= 66

9 . Evaluate √4356
A. 66 B. 55 C. 44 D. 33

Answer: A

Explanation:
By using prime factorization, we get
4356 = 2 x 2 x 3 x 3 x 11 x 11
√4356 = √(2 x 2 x 3 x 3 x 11 x 11)
= 2 × 3 × 11
Therefore, √4356 = 66.

10 . If √y/169 = 54/39, then y is equal to ?
A. 267 B. 324 C. 448 D. 527

Answer: B

Explanation:
√y/169= 54/39
⇒ y/169 = (54/39) x (54/39)
y= (54/39) x (54/39) x 169 = 324

11 . Evaluate √11025
A. 76 B. 92 C. 105 D. 162

Answer: C

Explanation:
By using prime factorization, we get
11025 = 5 x 5 x 3 x 3 x 7 x 7.
√11025 = √(5 x 5 x 3 x 3 x 7 x 7)
= 5 × 3 × 7
Therefore, √11025 = 105

12 . 112/√196 x √579/12 x √256/8 = ?
A. 17 B. 32 C. 53 D. 68

Answer: B

Explanation:
Given Expression 112/√196 x √579/12 x √256/8
=(112/14) x (24/12) x (16/8)
= 32

13 . √248 + √(52+ √144)
A. 16 B. 14 C. 11 D. 7

Answer: A

Explanation:
√248 + √(52 + √144)
= √248 + √(52 + 12)
=√248 + √64
= √248 + 8
= √256
= 16

14 . √176 + √2401 = ?
A. 25 B. 20 C. 15 D. 10

Answer: C

Explanation:
√176 + √2401
= √176 + 49
= √225 = 15

15 . Given that √4096 = 64, the value of √4096 + √40.96 + √.004096 is ?
A. 52.925 B. 67.251 C. 73.672 D. 70.464

Answer: D

Explanation:
√4096 + √40.96 + √.004096
= √4096 + √4096/100 + √4096/1000000
=√4096 + √4096/10 + √4096/1000
= 64 + 64/10 + 64/1000
= 70.464

16 . In an auditorium, the number of rows is equal to the number of chairs in each row. If the capacity of the auditorium is 2025, find the number of chairs in each row. A. 34 B. 45 C. 52 D. 66

Answer: B

Explanation:
Let the number of chairs in each row be x.
Then, the number of rows = x.
Total number of chairs in the auditorium = (x × x) = x²
But, the capacity of the auditorium = 2025 (given).
Therefore, x² = 2025
= 5 × 5 × 3 × 3 × 3 × 3
x = (5 × 3 × 3) = 45.
Hence, the number of chairs in each row = 45
17 . Find the smallest number by which 396 must be multiplied so that the product becomes a perfect square.
A. 11 B. 13 C. 15 D. 17

Answer: A

Explanation:
By prime factorization, we get
396 = 2 × 2 × 3 × 3 × 11
It is clear that in order to get a perfect square, one more 11 is required.
So, the given number should be multiplied by 11 to make the product a perfect square.

18 . √.04
A. 0.2 B. 0.7 C. 0.9 D. none of the above

Answer: A

Explanation:
√.04 = √4/100
= 2/10
= 0.2

19 . If √256 / √x = 2, then x is equal to ?
A. 54 B. 64 C. 74 D. 84

Answer: B

Explanation:
√256 / √x = 2
⇒ 16 = 2√x
⇒ √x = 8
∴ x = 64.

20 . √288 / √128
A. 9/2 B. 7/2 C. 5/2 D. 3/2

Answer: D

Explanation:
√288 / √128 = √9/4
= 3/2

21 . √10 x √15 ?
A. 4√3 B. 5√6 C. 7√7 D. 8√1

Answer: B

Explanation:
√10 x √15
= √10 x 15
= √150
= √25 x 6
= √25 x √6
= 5√6


22 . √ ? / 200 = 0.02 ?
A. 8 B. 12 C. 16 D. 22

Answer: C

Explanation:
Let √x / 200 = 0.02
Then, √x = 200 x 0.02 = 4.
So, x = 16

23 . Evaluate the cube root: ∛216
A. 15 B. 12 C. 9 D. 6

Answer: D
Explanation:
By prime factorization, we have
216 = 2 × 2 × 2 × 3 × 3 × 3
= (2 × 2 × 2) × (3 × 3 × 3)
Therefore, ∛216 = (2 × 3) = 6

24 . Evaluate the cube root: ∛343
A. 1 B. 5 C. 7 D. 11

Answer: C

Explanation:
By prime factorization, we have
343 = 7 × 7 × 7
= (7 × 7 × 7).
Therefore, ∛343 = 7

25 . 250 / √? = 10
A. 625 B. 560 C. 420 D. 286

Answer: A

Explanation:
Let 250 / √x = 10.
Then √x= 250 / 10 = 25
∴ x = 625

26 . √4375 / √7 = ?
A. 15 B. 25 C. 35 D. 55

Answer: B

Explanation:
√4375 / √7 = √4375 / 7
= √625
= 25

27 . Find the smallest number by which 1100 must be divided so that the quotient is a perfect square.
A. 40 B. 60 C. 80 D. 100

Answer: D

Explanation:
Expressing 1100 as the product of primes, we get
1100 = 2 × 2 × 5 × 5 × 11
Here, 2 and 5 occur in pairs and 11 does not.
Therefore, 1100 must be divided by 11 so that the quotient is 100
i.e., 1100 ÷ 11 = 100 and 100 is a perfect square.

28 . Evaluate the cube root: ∛2744
A. 11 B. 12 C. 13 D. 14

Answer: D

Explanation:
By prime factorization, we have
2744 = 2 × 2 × 2 × 7 × 7 × 7
= (2 × 2 × 2) × (7 × 7 × 7).
Therefore, ∛2744 = (2 × 7) = 14

29 . Find the cube root of (-1000). A. -10 B. 12 C. -16 D. 18

Answer: A

Explanation:
We know that ∛-1000 = -∛1000
Resolving 1000 into prime factors, we get
1000 = 2 × 2 × 2 × 5 × 5 × 5
= (2 × 2 × 2) × (5 × 5 × 5)
Therefore, ∛1000 = (2 × 5) = 10
Therefore, ∛-1000 = -(∛1000) = -10

30 . √4/3 - √3/4 = ?
A. 7/2√3 B. 3/2√3 C. 1/2√3 D. 5/2√3

Answer: C

Explanation:
√4/√3 - √3/√4
= 2/√3 - √3/2
= (4 - 3)/2√3
= 1/2√3