Class 8 RD Sharma Solutions – Chapter 4 Cubes and Cube Roots – Exercise 4.4 | Set 1

Question 1. Find the cube roots of each of the following integers:

(i) -125 

(ii) -5832

(iii) -2744000 

(iv) -753571

(v) -32768

Solution:

(i) -125 

The cube root of -125 will be ∛-125 

= -∛125 

= –∛5 × 5 × 5 

= -5

Hence, the cube root of -125 is -5

(ii) -5832

The cube root of -5832 will be ∛-5832 

= -∛2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 

= –∛2³ × 3³ × 3³ = -(2 × 3 × 3)

= -18

Hence, the cube root of -5832 is -18

(iii) -2744000

The cube root of -2744000 will be ∛-2744000

= -∛2 × 2 × 2 × 7 × 7 × 7 × 10 × 10 × 10 

= -∛2³ × 7³ × 10³ = -(2 × 7 × 10)

= -140

Hence, the cube root of -2744000 is -140

(iv) -753571

The cube root of –753571 will be ∛-753571

= -∛7 × 7 × 7 × 13 × 13 × 13 

= -∛7³ × 13³ 

= -(7 × 13)

= -91

Hence, the cube root of –753571 is -91

(v) -32768

The cube root of -32768 will be ∛-32768

= -∛2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= -∛2³ × 2³ × 2³ × 2³ × 2³ 

= -(2 × 2 × 2 × 2 × 2)

= -32

Hence, the cube root of –32768 is -32

Question 2. Show that:

(i) ∛27 × ∛64 = ∛(27 × 64)

(ii) ∛(64 × 729) = ∛64 × ∛729

(iii) ∛(-125 × 216) = ∛-125 × ∛216

(iv) ∛(-125 × -1000) = ∛-125 × ∛-1000

Solution:

(i) ∛27 × ∛64 = ∛(27 × 64)

In order to verify the equation, we must prove LHS = RHS

So, LHS = ∛27 × ∛64 

= ∛3 × 3 × 3 × ∛4 × 4 × 4

= ∛(3 × 3 × 3) × (4 × 4 × 4) = 3 × 4 = 12

 And, RHS = ∛(27 × 64)

= ∛(3 × 3 × 3 × 4 × 4 × 4) = 3 × 4 = 12

So, LHS = RHS = 12

Hence proved

(ii) ∛(64 × 729) = ∛64 × ∛729

In order to verify the equation, we must prove LHS = RHS

So, LHS = ∛ (64 × 729)

= ∛(4 × 4 × 4 × 9 × 9 × 9) = 4 × 9 = 36

And, RHS = ∛64 × ∛729

= ∛4 × 4 × 4 × ∛9 × 9 × 9

= ∛(4 × 4 × 4) × (9 × 9 × 9) = 4 × 9 = 36

So, LHS = RHS = 36

Hence proved

(iii) ∛(-125 × 216) = ∛-125 × ∛216

In order to verify the equation, we must prove LHS = RHS

So, LHS = ∛ (-125 × 216)

= ∛-5 × -5 × -5 × ∛2 × 2 × 2 × 3 × 3 × 3

= ∛(-5 × -5 × -5) × (2 × 2 × 2 × 3 × 3 × 3) 

= -5 × 2 × 3 = -30

And, RHS = ∛-125 × ∛216

= ∛((-5 × -5 × -5 × 2 × 2 × 2 × 3 × 3 × 3) 

= -5 × 2 × 3 = -30

So, LHS = RHS = -30

Hence proved

(iv) ∛(-125×-1000) = ∛-125 × ∛-1000

In order to verify the equation, we must prove LHS = RHS

So, LHS = ∛ (-125 × -1000)

= ∛-5 × -5 × -5 × ∛-10 × -10 × -10 

= ∛(-5 × -5 × -5) × (-10 × -10 × -10) = -5 × -10 = 50

And, RHS = ∛-125 × ∛-1000

= ∛((-5 × -5 × -5 × -10 × -10 × -10) 

= -5 × -10 = 50

So, LHS = RHS = 50

Hence proved

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

(i) 8 × 125  

(ii) -1728 × 216

(iii) -27 × 2744 

(iv) -729 × -15625

Solution:

(i) 8×125  

We know that ∛ab = ∛a × ∛b

So, ∛(8 × 125) = ∛8 × ∛125

= ∛(2 × 2 × 2) × ∛(5 × 5 × 5)

= 2 × 5 = 10

Hence, the cube root of 8 × 125 = 10

(ii) -1728 × 216

We know that ∛ab = ∛a × ∛b

So, ∛(-1728×216) = -∛1728 × ∛216

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

= -(2 × 2 × 3 × 2 × 3) = -72

Hence, the cube root of -1728 × 216 = -72

(iii) -27 × 2744 

We know that ∛ab = ∛a × ∛b

So, ∛(-27×2744) = -∛27 × ∛2744

= -∛(3 × 3 × 3) × ∛(2 × 2 × 2 × 7 × 7 × 7)

= -(3 × 2 × 7) = -42

Hence, the cube root of -27×2744 = -42

(iv) -729 × -15625

We know that ∛ab = ∛a × ∛b

So, ∛(-729 × -15625) = -∛729 × -∛15625

= -∛(3 × 3 × 3 × 3 × 3 × 3) × -∛(5 × 5 × 5 × 5 × 5 × 5)

= (3 × 3 × 5 × 5) = 225

Hence, the cube root of -729 × -15625 = 225

Question 4. Evaluate:

(i) ∛ (4³ × 6³)

(ii) ∛ (8 × 17 × 17 × 17)

(iii) ∛ (700 × 2 × 49 × 5)

(iv) 125 ∛a⁶ – ∛125a⁶

Solution:

(i) ∛(4³ × 6³)

We know that ∛(a × b) = ∛a × ∛b

So, ∛(4³ × 6³) = ∛4³ × ∛6³

= 4 × 6 = 24

Hence, ∛(4³ × 6³) = 24

(ii) ∛(8 × 17 × 17 × 17)

We know that ∛(a × b) = ∛a × ∛b

So, ∛ (8 × 17 × 17 × 17) = ∛8 × ∛17 × 17 × 17

= 2 × 17 = 34

Hence, ∛(8 × 17 × 17 × 17) = 34

(iii) ∛(700 × 2 × 49 × 5)

∛(700 × 2 × 49 × 5) 

= ∛ 2 × 2 × 5 × 5 × 7 × 2 × 7 × 7 × 5

= ∛ 2³ × 7³ × 5³

= 2 × 7 × 5 = 70 

Hence, ∛(700 × 2 × 49 × 5) = 70

(iv) 125 ∛a⁶ – ∛125a⁶

125 ∛a⁶ – ∛125a⁶ = 125∛(a²)³ – ∛5³ × (a²)³

= 125a² – 5a²

= 120a²

Hence, 125 ∛a⁶ – ∛125a⁶ = 120a²

Question 5. Find the cube root of each of the following rational numbers:

(i) -125/729

(ii) 10648/12167

(iii) -19683/24389

(iv) 686/-3456

(v) -39304/-42875

Solution :

(i) -125/729

We can write cube root of -125/729 as, 

∛-125/ ∛729 

= -(∛5 × 5 × 5)/(∛9 × 9 × 9)

= -5/9

Hence, the cube root of -125/729 is -5/9

(ii) 10648/12167

We can write cube root of 10648/12167 as, 

∛10648/ ∛12167 

= (∛2 × 2 × 2 × 11 × 11 × 11)/(∛23 × 23 × 23)

= (2 × 11)/ 23 = 22/23

Hence, the cube root of 10648/12167 is 22/23

(iii) -19683/24389

We can write cube root of -19683/24389 as, 

∛-19683/ ∛24389 

= -(∛3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3)/(∛29 × 29 × 29)

= -(3 × 3 × 3)/ 23 = -27/29

Hence, the cube root of –19683/24389 is -27/29

(iv) 686/-3456

We can write cube root of 686/-3456 as, 

∛686/ ∛-3456 

= -(∛2 × 7 × 7 × 7)/(∛2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3)

= -7/ (2 × 2 × 3) = -7/12

Hence, the cube root of 686/-3456 is -7/12

(v) -39304/-42875

We can write the cube root of -39304/-42875 as, 

∛-39304/ ∛-42875

= (∛2 × 2 × 2 × 17 × 17 × 17)/(∛5 × 5 × 5 × 7 × 7 × 7)

= (2 × 17)/ (5 × 7) = 34/35

Hence, the cube root of -39304/-42875 is 34/35

Question 6. Find the cube root of each of the following rational numbers:

(i) 0.001728

(ii) 0.003375

(iii) 0.001

(iv) 1.331

Solution :

(i) 0.001728 can be written as 1728/1000000

So cube root of 0.001728 can be derived as,

∛0.001728 = ∛1728 / ∛1000000

= ∛(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) / ∛(10 × 10 × 10 × 10 × 10 × 10)

= (2 × 2 × 3) / (10 × 10) = 12/100 

= 0.12

Hence, the cube root of 0.001728 is 0.12

(ii) 0.003375 can be written as 3375/1000000

So cube root of 0.003375 can be derived as,

∛0.003375 = ∛3375 / ∛1000000

= ∛(3 × 3 × 3 × 5 × 5 × 5) / ∛(10 × 10 × 10 × 10 × 10 × 10)

= (3 × 5) / (10 × 10) = 15/100 

= 0.15

Hence, the cube root of 0.003375 is 0.15

(iii) 0.001 can be written as 1/1000

So cube root of 0.001 can be derived as,

∛0.001 = ∛1 / ∛1000

= ∛(1 × 1 × 1) / ∛(10 × 10 × 10)

= 1/10 = 0.1

Hence, the cube root of 0.001 is 0.1

(iv) 1.331 can be written as 1331/1000

So cube root of 1.331 can be derived as,

∛1.331 = ∛1331 / ∛1000

= ∛(11 × 11 × 11) / ∛(10 × 10 × 10)

= 11/10 = 1.1

Hence, the cube root of 1.331 is 1.1

Question 7. Evaluate each of the following:

(i) ∛27 + ∛0.008 + ∛0.064

(ii) ∛1000 + ∛0.008 – ∛0.125

(iii) ∛(729/216) × 6/9

(iv) ∛(0.027/0.008) ÷ √(0.09/0.04) – 1

(v) ∛(0.1 × 0.1 × 0.1 × 13 × 13 × 13)

Solution:

(i) Simplifying ∛27 + ∛0.008 + ∛0.064 we get,

= ∛3 × 3 × 3 + ∛0.2 × 0.2 × 0.2 + ∛0.4 × 0.4 × 0.4 

= 3 + 0.2 + 0.4 = 3.6

Hence, ∛27 + ∛0.008 + ∛0.064 = 3.6

(ii) Simplifying ∛1000 + ∛0.008 – ∛0.125 we get,

= ∛10 × 10 × 10 + ∛0.2 × 0.2 × 0.2 – ∛0.5 × 0.5 × 0.5 

= 10 + 0.2 – 0.5 = 9.7

Hence, ∛1000 + ∛0.008 – ∛0.125 = 9.7

(iii) Simplifying ∛(729/216) × 6/9 we get,

= ∛(9 × 9 × 9)/(6 × 6 × 6) × 6/9 

= 9/6 × 6/9 = 1

Hence, ∛(729/216) × 6/9 = 1

(iv) Simplifying ∛(0.027/0.008) ÷ √(0.09/0.04) – 1 we get,

= ∛(0.3 × 0.3 × 0.3)/(0.2 × 0.2 × 0.2) ÷ √(0.3 × 0.3)/(0.2 × 0.2) – 1 

= 0.3/0.2 ÷ 0.3/0.2 – 1 = 1 – 1 = 0

Hence, ∛(0.027/0.008) ÷ √(0.09/0.04) – 1 = 0

(v) Simplifying ∛(0.1 × 0.1 × 0.1 × 13 × 13 × 13) we get,

= ∛(0.1 × 0.1 × 0.1 × 13 × 13 × 13) = 0.1 × 13

= 1.3

Hence, ∛(0.1 × 0.1 × 0.1 × 13 × 13 × 13) = 1.3

Question 8. Show that:

(i) ∛(729)/∛(1000) = ∛(729/1000)

(ii) ∛(-512)/∛(343) = ∛(-512/343)

Solution:

(i) ∛(729)/ ∛ (1000) = ∛(729/1000)

LHS = ∛(729)/ ∛(1000)

= ∛(9 × 9 × 9)/ ∛(10 × 10 × 10) = 9/10

RHS = ∛(729/1000)

= ∛(9 × 9 × 9/10 × 10 × 10) = 9/10

Since, LHS = RHS = 9/10

Hence, proved 

(ii) ∛(-512)/ ∛(343) = ∛(-512/343)

LHS = ∛(-512)/ ∛(343)

= -∛(8 × 8 × 8)/ ∛(11 × 11 × 11) = -8/11

RHS = ∛(-512/343)

= -∛(8 × 8 × 8/(11 × 11 × 11) = -8/11

Since, LHS = RHS = -8/11

Hence, proved