# Selina Concise Solutions for Chapter 4 Cubes and Cube Roots Class 8 ICSE Mathematics

**Exercise 4A**

**1. Find the cube of :**

**(i) 7**

**(ii) 11**

**(iii) 16**

**(iv) 23**

**(v) 31**

**(vi) 42**

**(vii) 54**

**Solution**

**(i)**(7)

^{3}= 7 × 7 × 7 = 343

**(ii)**(11)

^{3}= 11 × 11 × 11 = 1331

**(iii)**(16)

^{3}= 16 × 16 × 16 = 4096

**(iv)**(23)

^{3}= 23 × 23 × 23 = 12167

**(v)**(31)

^{3}= 31 × 31 × 31 = 29791

**(iv)**(42)

^{3}= 42 × 42 × 42 = 74088

**(vii)**(54)

^{3}= 54 × 54 × 54 = 157464

**2. Find which of the following are perfect cubes:**

**(i) 243**

**(ii) 588**

**(iii) 1331**

**(iv) 24000**

**(v) 1728**

**(vi) 1938**

**Solution**

**(i)**243

= (3 × 3 × 3) × 3

= 3

^{3}× 3

∴ 297 is not a perfect cube.

**(ii)**588

∴ 588 is not a perfect cube.

**1331**

(iii)

(iii)

^{3}

∴ 1331 is perfect cube.

**(iv)**24000

∵ 24000 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5 × 5 × 5

= (2)

^{3}× (2)

^{3}× (5)

^{3}× 3

∴ 24000 is not a perfect cube.

**(v)**1728

= (2)

^{3}× (2)

^{3}× (3)

^{3}

∴ 1728 is a perfect cube.

**(vi)**1938

1938 = 2 × 3 × 17 × 19

1938 is not a perfect cube.

**3. Find the cubes of :**

**(i) 2.1**

**(ii) 0.4**

**(iii) 1.6**

**(iv) 2.5**

**(v) 0.12**

**(vi) 0.02**

**(vii) 0.8**

**Solution**

**(i)**2.1

= (2.1)

^{3}= (21/10)

^{3}= (21 × 21 × 21)/(10 × 10 × 10)

= 9261/1000

= 9.261

= 9261/1000

= 9.261

**(ii)**0.4= (0.4)

^{3}= (4/10)

^{3}= (4 × 4 × 4)/(10 × 10 × 10)

= 64/1000

= 0.064

= 64/1000

= 0.064

**(iii)**1.6= (1.6)

^{3}= (16/10)

^{3}= (16 × 16 × 16)/(10 × 10 × 10)

= 4096/1000

= 4096/1000

= 4.096

**(iv)**2.5= (2.5)

^{3}= (25/10)

^{3}= (25 × 25 × 25)/(10 × 10 × 10)

= 15625/1000

= 15.625

= 15.625

**(v)**0.12= (0.12)

^{3}= (12/100)

^{3}= (12 × 12 × 12)/(100 × 100 × 100)

= 1728/1000000

= 1728/1000000

= 0.001728

**(vi)**0.02= (0.02)

^{3}= (2/100)

^{3}= (2 × 2 × 2)/(100 × 100 × 100)

= 8/1000000

= 8/1000000

= 0.000008

**(vii)**0.8= (0.8)

^{3}= (8/10)

^{3}= (8 × 8 × 8)/(10 × 10 × 10)

= 512/100

= 0.512

= 512/100

= 0.512

**4. Find the cubes of:****(i) 3/7****(ii) 8/9****(iii) 10/13****(iv) 1 2/7****(v) 2 ½**

**Solution****(i)**3/7= (3/7)

^{3}= (3 × 3 × 3)/(7 × 7 × 7)

= 27/343

**(ii)**8/9

= (8/9)

^{3}= (8 × 8 × 8)/(9 × 9 × 9)

= 512/729

**(iii)**10/13

= (10/13)

^{3}= (10 × 10 × 10)/(13 × 13 × 13)

= 1000/2197

**(iv)**1 2/7

= (1 2/7)

^{3}= {(1 × 7 + 2)/7}

^{3}= (9/7)

= (9 × 9 × 9)/(7 × 7 × 7)

^{3}= (9 × 9 × 9)/(7 × 7 × 7)

= 729/343

= 2 43/343

**(v)**2½

= (2 1/2)

^{3}= (5/2)

= (5 × 5 × 5)/(2 × 2 × 2)

^{3}= (5 × 5 × 5)/(2 × 2 × 2)

= 125/8

= 15 5/8

**5. find the cubes of :****(i) -3****(ii) -7****(iii) -12****(iv) -18****(v) -25****(vi) -30****(vii) -50**

**Solution****(i)**-3= (-3)

^{3}= -3 × -3 × -3

= -(3 × 3 × 3)

= -(3 × 3 × 3)

= -27

**(ii)**-7

= (-7)

= -7 × -7 × -7

= -(7 × 7 × 7)

= -(7 × 7 × 7)

= -343

**(iii)**-12

= (-12)

^{3}= -12 × -12 × -12

= -(12 × 12 ×12)

= -(12 × 12 ×12)

= -1728

**(iv)**-18

= (-18)

^{3}= -18 × -18 × -18

= -(18 × 18 × 18)

= -(18 × 18 × 18)

= -5832

**(v)**-25

= (-25)

^{3}= -25 × -25 × -25

= -(25 × 25 × 25)

= -(25 × 25 × 25)

= -15625

**(vi)**-30

= (-30)

^{3}= -30 × -30 × -30

= -(30 × 30 × 30)

= -(30 × 30 × 30)

= -27000

**(vii)**-50

= (-50)

^{3}= -50 × -50 × -50

= -(50 × 50 × 50)

= -(50 × 50 × 50)

= -125000

**6. Which of the following are cubes of:****(i) an even number****(ii) an odd number****216, 729, 3375, 8000, 125, 343, 4096 and 9261.**

**Solution**

∵ 216 = 2 × 2 × 2 × 3 × 3 × 3

^{3}× (3)

^{3}= (6)

^{3}

∵ 729 = 3 × 3 × 3 × 3 × 3 × 3

= (3)

^{3}× (3)^{3}= (9)^{3}
∵ 3375 = 5 × 5 × 5 × 3 × 3 × 3

= (5)

^{3}× (3)^{3}= (15)^{3}
∵ 8000 = 20 × 20 × 20 = (20)

125 = 5 × 5 × 5 = (5)

∵ 343 = 7 × 7 × 7 = (7)

∵ 4096 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= (2)

(i) Cubes of an even number are 216, 8000, 4096.

(ii) Cubes of an odd number are 729, 3375, 125, 343, 9261.

^{3}^{3}^{3}∵ 4096 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

^{3}× (2)^{3}× (2)^{3}× (2)^{3}= (16)^{3}(i) Cubes of an even number are 216, 8000, 4096.

(ii) Cubes of an odd number are 729, 3375, 125, 343, 9261.

**7. Find the least number by which 1323 must be multiplied so that the product is a perfect cube.**

**Solution**The prime factor of 1323 are = 3 × 3 × 3 × 7 × 7

= (3 × 3 × 3) × 7 × 7

Clearly, 1323 must be multiplied by 7.

= (3 × 3 × 3) × 7 × 7

Clearly, 1323 must be multiplied by 7.

**8. Find the smallest number by which 8768 must be divided so that the quotient is a perfect cube.**

**Solution**The prime factor 8768 are

= 2 × 2 × 2 × 2 × 2 × 2 × 137

= (2 × 2 × 2) × (2 × 2 × 2) × 137

Clearly, 8768 must be divided by 137.

= 3 × 3 × 3 × 3 × 7 × 7 × 7

= (3 × 3 × 3) × (7 × 7 × 7) × 3

Clearly, 27783 must be multiplied by 3 × 3 = 9

= (2 × 2 × 2) × (2 × 2 × 2) × 137

Clearly, 8768 must be divided by 137.

**9. Find the smallest number by which 27783 be multiplied to get a perfect square number.**

**Solution**= 3 × 3 × 3 × 3 × 7 × 7 × 7

= (3 × 3 × 3) × (7 × 7 × 7) × 3

Clearly, 27783 must be multiplied by 3 × 3 = 9

**10. With what least number must 8640 be divided so that the quotient is a perfect cube?**

**Solution**The prime factor of 8640 are

= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5

= (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) × 5

Clearly, 8640 must be divided by 5.

The prime factors of 77175 are

= 3 × 3 × 5 × 5 × 7 × 7 × 7

= (7 × 7 × 7) × 3 × 3 × 5 × 5

Clearly, 77175 must be multiplied by 3 × 5 = 15

= (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) × 5

Clearly, 8640 must be divided by 5.

**11. Which is the smallest number that must be multiplied to 77175 to make it a perfect cube?**

**Solution**The prime factors of 77175 are

= (7 × 7 × 7) × 3 × 3 × 5 × 5

Clearly, 77175 must be multiplied by 3 × 5 = 15

### Exercise 4 B

**1. Find the cube-roots of :**

**(i) 64**

**(ii) 343**

**(iii) 729**

**(iv) 1728**

**(v) 9261**

**(vii) 4096**

**(vii) 8000**

**(viii) 3375**

**Solution**

**(i)**

**(ii)**

**(iii)**729 = ∛729 = (3 × 3 × 3) × (3 × 3 × 3)

= 3 × 3

= 9

**(iv)**

**(v)**

**(vi)**

**(vii)**

**(viii)**

**2. Find the cube-roots of :**

**(i) 27/64**

**(ii) 125/216**

**(iii) 343/512**

**(iv) 64 × 729**

**(v) 64 × 27**

**(vi) 729 × 8000**

**(vii) 3375 × 512**

**Solution**

**(i)**

**(ii)**

**(iii)**

**(iv)**

**(v)**

**(vi)**

**(vii)**

**3.**

**Find the cube-roots of :**

**(i) -216**

**(ii) -512**

**(iii) -1331**

**(iv) -27/125**

**(v) -64/343**

**(vi) -512/343**

**(vii) -2197**

**(viii) -5832**

**(ix) -2744000**

**Solution**

**(i)**

**(ii)**

**(iii)**

**(iv)**

**(v)**

**(vi)**

**(vii)**

**(viii)**

**(ix)**

**4. Find the cube-roots of :**

**(i) 2.744**

**(ii) 9.261**

**(iii) 0.000027**

**(iv) -0.512**

**(v) -15.625**

**(vi) -125 x 1000**

**Solution**

**(i)**

**(ii)**

**(iii)**

**(iv)**

**(v)**

**(vi)**

**5. Find the smallest number by which 26244 may be divided so that the quotient is a perfect cube.**

**Solution**

The prime factors of 26244 are

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

= (3 × 3 × 3) × (3 × 3 × 3) × 3 × 3 × 2 × 2

Clearly, 26244 must be divided by 3 × 3 × 2 × 2 = 36

**6. What is the least number by which 30375 should be multiplied to get a perfect cube?**

**Solution**

The prime factors of 30375 are

= 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5

= (3 × 3 × 3) × (5 × 5 × 5) × 3 × 3

clearly, 30375 must be multiplied with 3

**7. Find the cube-roots of :**

**(i) 700 x 2 x 49 x 5**

**(ii) -216 x 1728**

**(iii) -64 x -125**

**(iv) -27/343**

**(v) 729/-1331**

**(vi) 250.047**

**(vii) -175616**

**Solution**

**(i)**700 × 2 × 49 × 5

= 2 × 2 5 × 5 × 7 × 2 × 7 × 7 × 5

= (2 × 2 × 2) × (5 × 5 × 5) × (7 × 7 × 7)

= 2 × 5 × 10 = 70

**(ii)**-216 × 1728

= -(2 × 2 × 2 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3)

= -2 × 3 × 2 × 2 × 2 × 3

= -72

**(iii)**-64 × -125

= -(4 × 4 × 4) × -(5 × 5 × 5)

= -4 × -5

= 20

**(iv)**- 27/343

= (3 × 3 × 3)/(7 × 7 × 7)

= -3/7

**(v)**729/-1331

= (9 × 9 × 9)/-(11 × 11 ×11)

= - 9/11

**(vi)**250.047 = 250047/1000

= {(3 × 3 × 3) × (3 × 3 × 3) × (7 × 7 × 7)/(10 × 10 × 10)}

= (3 × 3 × 7)/10

= 63/10

= 6.3

**(vii)**-175616

= -[(2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (7 × 7 × 7)]

= -[2 × 2 × 2 × 7]

= -56