# Cubes And Cube Roots

## Exercise 6.1

1). Which of the following are not perfect cubes?

(i) 216 (ii) 128

(iii) 1000 (iv) 100

(v) 46656

Solution:

(i) 216

216 = 2 X 2 X 2 X 3 X 3 X 3

216 = __2 X 2 X 2__ X __3 X 3 X 3 __

Group of factors 2 and 3 is formed

Therefore, 216 is a perfect cube.

(ii) 128

128 = 2 X 2 X 2 X 2 X 2 X 2 X 2

128 = __2 X 2 X 2__ X __2 X 2 X 2__ X 2

in this factorisation, factor 2 cannot be formed in groups of 3 Therefore, 128 is not a perfect cube.

(iii) 1000

1000 = 2 X 2 X 2 X 5 X 5 X 5

1000 = __2 X 2 X 2__ X __5 X 5 X 5 __

in this factorisation, group of three of factors 2 and 5 is formed

Therefore, 1000 is a perfect cube.

(iv) 100

100 = 2 X 2 X 5 X 5

in this factorisation, factors 2 and 5cannot be formed in groups of 3

Therefore, 100 is not a perfect cube.

(v) 46656

46656 = 2 X 2 X 2 X 2 X 2 X 2 X 3 X 3 X 3 X 3 X 3 X 3

46656 = __2 X 2 X 2__ X __2 X 2 X 2__ X __3 X 3 X 3__ X __3 X 3 X 3__

in this factorisation, group of three of factors 2 and 3 formed therefore, 46656 is a perfect cube.

2). Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.

(i) 243 (ii) 256

(iii) 72 (iv) 675

(v) 100

Solution:

(i) 243

243 = 3 X 3 X 3 X 3 X 3

243 = __3 X 3 X 3__ X 3 X 3

Here one 3 is required to make the group of three

The least number multiplied to obtain a perfect cube = 3

(ii) 256

256 = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2

256 = __2 X 2 X 2__ X __2 X 2 X 2__ X 2 X 2

Here one 2 is required to make the group of three

The least number multiplied to obtain a perfect cube = 2

(iii) 72

72 = 2 X 2 X 2 X 3 X 3

72 = __2 X 2 X 2__ X 3 X 3

Here one 3 is required to make the group of three

The least number multiplied to obtain a perfect cube = 3

(iv) 675

675 = 3 X 3 X 3 X 5 X 5

675 = __3 X 3 X 3__ X 5 X 5

Here one 5 is required to make the group of three

The least number multiplied to obtain a perfect cube = 5

(v) 100

100 = 2 X 2 X 5 X 5

Here one 2 and one 5 is required to make the group of three

The least number multiplied to obtain a perfect cube = 2 X 5

= 10

3). Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

(i) 81 (ii) 128

(iii) 135 (iv) 192

(v) 704

Solution:

(i) 81

81 = 3 X 3 X 3 X 3

81 = __3 X 3 X 3__ X 3

Here 3 is the number by which 81 is divided to get a perfect cube

The least number to be divided to obtain a perfect cube = 3

(ii) 128

128 = 2 X 2 X 2 X 2 X 2 X 2 X 2

128 = __2 X 2 X 2__ X __2 X 2 X 2__ X 2

Here 2 is the number by which 128 is divided to get a perfect cube

The least number to be divided to obtain a perfect cube = 2

(iii) 135

135 = 3 X 3 X 3 X 5

135 = __3 X 3 X 3__ X 5

Here 5 is the number by which 135 is divided to get a perfect cube

The least number to be divided to obtain a perfect cube = 5

(iv) 192

192 = 2 X 2 X 2 X 2 X 2 X 2 X 3

192 = __2 X 2 X 2__ X __2 X 2 X 2__ X 3

Here 3 is the number by which 192 is divided to get a perfect cube

The least number to be divided to obtain a perfect cube = 3

(v) 704

704 = 2 X 2 X 2 X 2 X 2 X 2 X 11

704 = __2 X 2 X 2__ X __2 X 2 X 2__ X 11

Here 11 is the number by which 704 is divided to get a perfect cube

The least number to be divided to obtain a perfect cube = 11

4). Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. how much such cuboids will he need to form a cube?

Sides of cuboid of plasticine are 5 cm, 2 cm and 5 cm

Volume of cuboid = 5 X 5 X 2

To make it perfect cube we have to make a group of three of factors 5 and 2.

Cuboids need to form a cube = 5 X 2 X 2

= 20

Click here for the solutions of Std 8 Maths

1). Rational Numbers

2). Linear Equations in One Variable

3). Understanding Quadrilaterals

4). Data Handling

5). Squares and Square Roots

6). Cubes and Cube Roots