**Question 1. Find the cubes of:**

**(i) -11**

**(ii) -12**

**(iii) -21**

**Solution:**

i)Cube of -11 = (-11)^{3}= -11 × -11 × -11 = -1331

ii)Cube of -12 = (-12)^{3}= -12 × -12 ×-12 = -1728

iii)Cube of -21 = (-21)^{3}= -21 × -21 ×-21 = -9261

**Question 2. Which of the following numbers are cubes of negative integers.**

**(i) -64**

**(ii) -1056**

**(iii) -2197**

**(iv) -2744**

**(v) -42875**

**Solution:**

In order to find out there the given negative number is a perfect cube or not, we need to check if its corresponding positive number is a perfect cube.

**i) -64**

Let’s first check whether 64 is a perfect cube or not.

Prime factorization of 64

64 = 2 × 2 × 2 × 2 × 2 × 2

Also, 64 = (2 × 2 × 2) × (2 × 2 × 2)

Since, 64 can be completely grouped in triplets of the equal factors,

So, 64 is a perfect cube of 4.

Hence, -64 is a perfect cube of negative number i.e -4.

**ii) -1056**

Let’s first check whether 1056 is a perfect cube or not.

Prime factorization of 1056

1056 = 2 × 2 × 2 × 2 × 2 × 3 × 11

Also, 1056 = (2 × 2 × 2) × 2 × 2 × 3 × 11

Since, 1056 can’t be completely grouped in triplets of the equal factors,

So, 1056 is not a perfect cube.

Hence, -1056 is a not perfect cube of a negative number.

**iii) -2197**

Let’s first check whether 2197 is a perfect cube or not.

Prime factorization of 2197

2197 = 13 × 13 × 13

Also, 2197 = (13 × 13 × 13)

Since, 2197 can be completely grouped in triplets of the equal factors,

So, 2197 is a perfect cube of 13.

Hence, -2197 is a perfect cube of negative number i.e -13.

**iv) -2744**

Let’s first check whether 2744 is a perfect cube or not.

Prime factorization of 2744

2744 = 2 × 2 × 2 × 7 × 7 × 7

Also, 2744 = (2 × 2 × 2) × (7 × 7 × 7)

Since, 2744 can be completely grouped in triplets of the equal factors,

So, 2744 is a perfect cube of 14.

Hence, -2744 is a perfect cube of negative number i.e -14.

**v) -42875**

Let’s first check whether 42875 is a perfect cube or not.

Prime factorization of 42875

42875 = 5 × 5 × 5 × 7 × 7 × 7

Also, 42875 = (5 × 5 × 5) × (7 × 7 × 7)

Since, 42875 can be completely grouped in triplets of the equal factors,

So, 42875 is a perfect cube of 35.

Hence, -42875 is a perfect cube of negative number i.e -35.

**Question 3. Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer :**

**(i) -5832 (ii) -2744000**

**Solution:**

**i) -5832**

Let’s first check whether 5832 is a perfect cube or not.

Prime factorization of 5832

5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

Also, 5832 = (2 × 2 × 2) × (3 × 3 × 3) × (3 × 3 × 3)

Since, 5832 can be completely grouped in triplets of the equal factors,

So, 5832 is a perfect cube of 18.

Hence, -5832 is a perfect cube of negative number i.e -18.

**ii) 2744000**

Let’s first check whether 2744000 is a perfect cube or not.

Prime factorization of 2744000

2744000 = 2 × 2 × 2 × 7 × 7 × 7 × 2 × 2 × 2 × 5 × 5 × 5

Also, 2744000 = (2 × 2 × 2) × (7 × 7 × 7) × (2 × 2 × 2) × (5 × 5 × 5)

Since, 2744000 can be completely grouped in triplets of the equal factors,

So, 2744000 is a perfect cube of 140.

Hence, -2744000 is a perfect cube of negative number i.e -140.

**Question 4. Find the cube of :**

**(i) 7/9**

**(ii) -8/11**

**(iii) 12/7**

**(iv) -13/8**

**(v) 12/5**

**(vi) 13/4**

**(vii) 0.3**

**(vii) 1/5**

**(ix) 0.08**

**(x) 2.1**

**Solution:**

i)Cube of 7/9 will be 7/9 × 7/9 × 7/9= 343/729

Hence, the cube of 7/9 is 343/729

ii)Cube of -8/11 will be -8/11 × -8/11 × -8/11= -512/1331

Hence, the cube of -8/11 is -512/1331

iii)Cube of 12/7 will be 12/7 × 12/7 × 12/7= 1728/343

Hence, the cube of 12/7 is 1728/343

iv)Cube of -13/8 will be -13/8 × -13/8 × -13/8= -2197/512

Hence, the cube of -13/8 is -2197/512

v)Cube of 12/5 will be 12/5 × 12/5 × 12/5= 1728/125

Hence, the cube of 12/5 is 1728/125

vi)Cube of 13/4 will be 13/4 × 13/4 × 13/4= 2197/64

Hence, the cube of 13/4 is 2197/64

vii)0.3 = 3/10So, Cube of 3/10 will be 3/10 × 3/10 × 3/10

= 27/1000 = 0.027

Hence, the cube of 0.3 is 0.027

viii)1.5 = 15/10So, Cube of 15/10 will be 15/10 × 15/10 × 15/10

= 3375/1000 = 3.375

Hence, the cube of 1.5 is 3.375

ix)0.08 = 8/100So, Cube of 8/100 will be 8/100 × 8/100 × 8/100

= 512/1000000 = 0.000512

Hence, the cube of 0.08 is 0.000512

x)2.1 = 21/10So, Cube of 21/10 will be 21/10 × 21/10 × 21/10

= 9261/1000 = 9.261

Hence, the cube of 2.1 is 9.261

**Question 5. Which of the following numbers are cubes of rational numbers :**

**(i) 27/64**

**(ii) 125/128**

**(iii) 0.001331**

**(iv) 0.04**

**Solution:**

**i) 27/64**

Factorization of 27/64 will be

27/64 = (3 × 3 × 3)/(4 × 4 × 4) = (3/4)

^{3}It means the cube of 3/4 is 27/64

Hence, we can say that 27/64 is a cube of rational number i.e 3/4

**ii) 125/128**

Factorization of 125/128 will be

125/128 = (5 × 5 × 5)/(2 × 2 × 2) × (2 × 2 × 2) × 2 = 5

^{3}/2^{3}× 2^{3 }× 2It means 125/128 is not perfect cube

Hence, we can say that 125/158 is not a cube of rational number

**iii)** **0.001331**

0.001331 = 1331/1000000

Factorization of 1331/1000000 will be

1331/1000000 = (11 × 11 × 11)/(10 × 10 × 10) × (10 × 10 × 10) =11

^{3}/10^{3}× 10^{3}It means the cube of 11/100 is 1331/1000000

Hence, we can say that 0.001331 is a cube of rational number i.e 0.11

**iv) 0.04**

0.04 = 4/100

Factorization of 4/100 = 2 × 2/ 10 × 10

It means 4/100 is not a perfect cube

Hence, we can say that 0.04 is not a cube of a rational number