ML Aggarwal: Cubes & Roots - 2
Q.1. Find the cube root of each of the following numbers by prime factorization:
(i) 12167
(ii) 35937
(iii) 42875
(iv) 21952
(v) 373248
(vi) 32768
(vii) 262144
(viii) 157464
(i) 12167 = 
Thus, the cube root of 12167 is 23.
(ii) 35937 = 
= 3 × 11 = 33
Thus, the cube root of 35937 is 33.
(iii) 42875 = 
=
= 3 × 7 = 35
Thus, the cube root of 42875 is 35.
(iv) 21952 = 
= 2 × 2 × 7 = 28
Thus, the cube root of 21952 is 28.
(v) 373248
= 2 × 2 × 2 × 3 × 3 = 72
Thus, the cube root of 373248 is 72.
(vi) 32768 = 
=
= 2 × 2 × 2 × 2 × 2 = 32
Thus, the cube root of 32768 is 32
(vii) 262144 = 
=
= 2 × 2 × 2 × 2 × 2 × 2 = 64
Thus, the cube root of 262144 is 64.
(viii) 157464 = 
=
= 2 × 3 × 3 × 3 = 54
Thus, the cube root of 157464 is 54.
Q.2. Find the cube root of each of the following cube numbers through estimation.
(i) 19683
(ii) 59319
(iii) 85184
(iv) 148877
(i) 19683
Grouping in 3's from right to left, we have 19,683
In the first group, 683 the unit digit is 3
So, the cube root will end with 7 and in the second group, 19
Cubing 23 = 8 and 33 = 27 As, 8 < 19 < 27
The ten's digit of the cube root will be 2 Thus, the cube root of 19683 is 27.
(ii) 59319
grouping in 3's from right to left, we have 59,319
In first group 319, and unit digit is 9
So, the unit digit of its cube root will be 9 and in the second group, 59
Cubing 33 = 27 and 43 = 64 As, 27 < 59 < 64
The ten's digit of the cube root will be 3 Thus, the cube root of 59319 is 39.
(iii) 85184
grouping in 3's from right to left, we have 85,184
In the first group 184, the unit digit is 4
So, the unit digit of its cube root will be 4 and in the second group, 85
Cubing 43 = 64 and 53 = 125
As, 64 < 85 < 125
ten's digit of cube root will be 4 Thus, the cube root of 85184 is 44.
(iv) 148877
Grouping in 3's, from right to left, we have 148,877
In the first group 877, unit digit is 7
So, the unit digit of cube root will be 3 and in the second group, 148
Cubing 53 = 125, 63 = 216 125 < 148 < 216
The ten's digit of cube root will be 5
Thus, the cube root of 148877 is 53.
Q.3. Find the cube root of each of the following numbers:
(i) -250047
(ii) 
(iii) 
(iv) 
(i) 
Thus, the cube root of -250047 is -63.
(ii) 
Performing prime factorization for both the numerator and denominator, we have
∴
(iii) 
=
∴
(iv) 
=
=
∴
Q.4. Evaluate the following:
(i) 
=
=
= (2) × 2) × (2) × (3) × (3) = 72
∴
(ii) 
Q.5. Find the cube root of the following decimal numbers:
(i) 0.003375
(ii) 19.683
(i) 0.003375
=
(ii) 19.683
=
=
Q.6.
= 3 + 0.2 + 0.4 = 3.6
Q.7. Multiply 6561 by the smallest number so that product is a perfect cube. Also, find the cube root of the product.
Performing prime factorization of 6561,
we get 6561 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3
6561 = (3 × 3 × 3) × (3 × 3 × 3) × 3 × 3
After grouping of the equal factors in 3's, it's seen that 3 × 3 is left ungrouped in 3's.
In order to complete it in triplet, we should multiply it by 3.
Hence, required smallest number = 3
and cube root of the product = 3 × 3 × 3 = 27
Q.8. Divide the number 8748 by the smallest number so that the quotient is a perfect cube. Also, find the cube root of the quotient.
Given number is 8748
On prime factorizing,
we get 8748 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 × 3
Grouping of the equal factor in 3's, it's seen that 2 × 2 × 3 is left without grouping.
8748 = 2 × 2 × 3 × (3 × 3 × 3) × (3 × 3 × 3)
Hence, on dividing the number 8748 by 12, we get 729
And, the cube root of 729 is 3 × 3 = 9.
Q.9. The volume of a cubical box is 21952 m3. Find the length of the side of the box.
Given, the volume of a cubical box is 21952 m2.
We know that,
Its edge =
=
= 2 × 2 × 7 = 28 m
Thus, the length of the side of the box is 28 m.
Q.10. Three numbers are in the ratio 3 : 4 : 5. If their product is 480, find the numbers.
Given,
Three numbers are in the ratio 3:4:5 and their product = 480
let's assume the numbers to be 3x, 4x and 5x, then we have
3x × 4x × 5x = 480
⇒ 60x3 = 480
⇒ x3 = 480/60 = 8 = 23
x = 2
Thus, the number are 2 × 3, 2 × 4 and 2 × 5 = 6, 8 and 10.
Q.11. Two numbers are in the ratio 4: 5. If difference of their cubes is 61, find the numbers.
Given,
Two numbers are in the ratio = 4 : 5
Difference between their cubes = 61
Let's assume the numbers to be 4x and 5x
So, we have
(5x)3 - (4x)3 = 61
125x3 - 64x3 = 61
61x3 = 61
⇒ x3 = 1 = (1)3
∴ x = 1
Hence, 4x = 4 × 1 = 4 and 5x = 5 × 1 = 5
Therefore, the numbers are 4 and 5
Q.12. Difference of two perfect cubes is 387. If the cube root of the greater of two numbers is 8, find the cube root of the smaller number.
Given,
The difference in two cubes = 387
And, the cube root of the greater number = 8
So, the greater number = (8)3 = 8 × 8 × 8 = 512
Hence, the second number = 512 - 387 = 125
Thus, The cube root of 125 is
