## 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 2^{3} = 8 and 3^{3} = 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

3^{3} = 27, 4^{3} = 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,

4^{3} = 64 and 5^{3} = 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

5^{3} = 125, 6^{3} = 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 m^{3}. Find the length of the side of the box.

Solution:

The volume of a cubical box 21952 m^{2}

= \(\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 7} \mathrm{m}\)

= 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

⇒ x^{3} = \(\frac{480}{60}\) = 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

125x^{3} – 64x^{3} = 61 ⇒ 61x^{3} = 61

⇒ x^{3} = 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