## ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 4 Cubes and Cube Roots Ex 4.2

Question 1.
Find the cube root of each of the following numbers by prime factorisation:
(i) 12167
(ii) 35937
(iii) 42875
(iv) 21952
(v) 373248
(vi) 32768
(vii) 262144
(viii) 157464
Solution:

Question 2.
Find the cube root of each of the following cube numbers through estimation.
(i) 19683
(ii) 59319
(iii) 85184
(iv) 148877
Solution:
(i) 19683
Grouping in 3’s from right to left, 19,683
In the first group 683, the unit digit is 3
∴ The cube root will be 7
and in the second group, 19
Cubing 23 = 8 and 33 = 27
∴ 8 < 14 < 21
∴ The tens digit of the cube will be 2
∴ Cube root of 19683 = 27

(ii) 59319
Grouping in 3’s, from right to left. 59,319
In first group, 319 unit digit is 9
∴ Unit digit of its cube root will be 9
and group 2nd, 59
33 = 27, 43 = 64
27 < 59 < 64
∴ Ten’s digit will be 3
∴ Cube root = 39

(iii) 85184
Grouping in 3’s from right to left 85,184
In group first 184, the unit digit is 4
∴ Unit digit of its cube root will be 4 and in group 2nd 85,
43 = 64 and 53 = 125
64 < 85 < 125
∴ Ten’s digit of cube root will be 4
∴ Cube root = 44

(iv) 148877
Grouping in 3’s, from right to left 148,877
In the first group 877, unit digit is 7
∴ The unit digit of cube root will be 3
and in group 2nd 148
53 = 125, 63 = 216
125 < 148 < 216
∴ Ten’s digit of cube root will be 5
∴ Cube root = 53

Question 3.
Find the cube root of each of the following numbers:

Solution:

Question 4.
Evaluate the following:
(i) $$\sqrt[3]{512 \times 729}$$
(ii) $$\sqrt[3]{(-1331) \times(3375)}$$
Solution:

Question 5.
Find the cube root of the following decimal numbers:
(i) 0.003375
(ii) 19.683
Solution:

Question 6.
Evaluate: $$\sqrt[3]{27}+\sqrt[3]{0.008}+\sqrt[3]{0.064}$$
Solution:

Question 7.
Multiply 6561 by the smallest number so that product is a perfect cube. Also, find the cube root of the product.
Solution:
6561
Factorising, we get

6561 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3
Grouping of the equal factors in 3’s,
we see 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

Question 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.
Solution:
8748
Factorising, we get

8748 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
Grouping of the equal factor in 3’s
we get that 2 × 2 × 3 is left without grouping.
So, dividing 8748 by 12, we get 729
whose cube root is 3 × 3 = 9

Question 9.
The volume of a cubical box is 21952 m3. Find the length of the side of the box.
Solution:
The volume of a cubical box 21952 m2

= $$\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 7} \mathrm{m}$$
= 2 × 2 × 7 = 28 m

Question 10.
Three numbers are in the ratio 3 : 4 : 5. If their product is 480, find the numbers.
Solution:
Three numbers are in the ratio 3:4:5
and their product = 480
Let numbers be 3x, Ax, 5x, then
3x × 4x × 5x = 480 ⇒ 60 × 3 = 480
⇒ x3 = $$\frac{480}{60}$$ = 8 = (2)3
∴ x = 2
∴ Number are 2 × 3, 2 × 4, 2 × 5
= 6, 8 and 10

Question 11.
Tw’o numbers are in the ratio 4 : 5. If difference of their cubes is 61, find the numbers.
Solution:
Two numbers are in the ratio = 4 : 5
Difference between their cubes = 61
Let the numbers be 4x, 5x
∴ (5x)3 – (4x)3 = 61
125x3 – 64x3 = 61 ⇒ 61x3 = 61
⇒ x3 = 1 = (1)3
∴ x = 1
∴ Numbers are 4x = 4 × 1 = 4 and 5 × 1 = 5
Hence numbers are = 4, 5

Question 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.
Solution:
Difference in two cubes = 387
Cube root of the greater number = 8
∴ Greater number = (8)3 = 8 × 8 × 8 = 512
Hence, second number = 512 – 387 = 125
and cube root of 125 = $$\sqrt[3]{125}$$
= $$\sqrt[3]{5 \times 5 \times 5}$$ = 5