# Class 8 RD Sharma Solutions- Chapter 4 Cubes and Cube Roots – Exercise 4.3

• Last Updated : 26 Nov, 2020

### (iii) 1728

Solution:

(i) 64

Attention reader! All those who say programming isn't for kids, just haven't met the right mentors yet. Join the  Demo Class for First Step to Coding Coursespecifically designed for students of class 8 to 12.

The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future.

Performing successive subtraction:

64 – 1 = 63

63 – 7 = 56

56 – 19 =37

37 – 37 = 0

Since subtraction is performed 4 times.

Therefore, the cube root of 64 is 4.

(ii) 512

Performing successive subtraction:

512 – 1 = 511

511 – 7 = 504

504 – 19 = 485

485 – 37 = 448

448 – 61 = 387

387 – 91 = 296

296 – 127 = 169

169 – 169 = 0

Since subtraction is performed 8 times.

Therefore, the cube root of 512 is 8.

(iii) 1728

Performing successive subtraction:

1728 – 1 = 1727

1727 – 7 = 1720

1720 – 19 = 1701

1701 – 37 = 1664

1664 – 91 = 1512

1512 – 127 = 1385

1385 – 169 = 1216

1216 – 217 = 999

999 – 271 = 728

728 – 331 = 397

397 – 397 = 0

Since subtraction is performed 12 times.

Therefore, the cube root of 1728 is 12.

### (iv) 1331

Solution:

(i) 130

Performing successive subtraction:

130 – 1 = 129

129 – 7 = 122

122 – 19 = 103

103 – 37 = 66

66 – 61 = 5

Since the next number to be subtracted is 91, which is greater than 5

Therefore,130 is not a perfect cube.

(ii) 345

Performing successive subtraction:

345 – 1 = 344

344 – 7 = 337

337 – 19 = 318

318 – 37 = 281

281 – 61 = 220

220 – 91 = 129

129 – 127 = 2

Since the next number to be subtracted is 169, which is greater than 2

Therefore, 345 is not a perfect cube

(iii) 792

Performing successive subtraction:

792 – 1 = 791

791 – 7 = 784

784 – 19 = 765

765 – 37 = 728

728 – 61 = 667

667 – 91 = 576

576 – 127 = 449

449 – 169 = 280

280 – 217 = 63

Since the next number to be subtracted is 271, which is greater than 63

Therefore, 792 is not a perfect cube

(iv) 1331

Performing successive subtraction:

1331 – 1 = 1330

1330 – 7 = 1323

1323 – 19 = 1304

1304 – 37 = 1267

1267 – 61 = 1206

1206 – 91 = 1115

1115 – 127 = 988

988 – 169 = 819

819 – 217 = 602

602 – 271 = 331

331 – 331 = 0

Since subtraction is performed 11 times,

Therefore, Cube root of 1331 is 11

Hence, 1331 is a perfect cube.

### Question 3: Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes, to make them perfect cubes. What are the corresponding cube roots?

Solution:

In the previous question, there are three numbers that are not perfect cubes.

(i) 130

Performing successive subtraction:

130 – 1 = 129

129 – 7 = 122

122 – 19 = 103

103 – 37 = 66

66 – 61 = 5

The next number which is to be subtracted is 91, which is greater than 5

Since, 130 is not a perfect cube.

Therefore, to make it a perfect cube we have to subtract 5.

130 – 5 = 125

125 is a perfect cube of 5.

(ii) 345

Performing successive subtraction:

345 – 1 = 344

344 – 7 = 337

337 – 19 = 318

318 – 37 = 281

281 – 61 = 220

220 – 91 = 129

129 – 127 = 2

The next number which is to be subtracted is 169, which is greater than 2

Since, 345 is not a perfect cube.

Therefore, to make it a perfect cube we have to subtract 2.

345 – 2 = 343

343 is a perfect cube of 7.

(iii) 792

Performing successive subtraction:

792 – 1 = 791

791 – 7 = 784

784 – 19 = 765

765 – 37 = 728

728 – 61 = 667

667 – 91 = 576

576 – 127 = 449

449 – 169 = 280

280 – 217 = 63

The next number which is to be subtracted is 271, which is greater than 63

Since, 792 is not a perfect cube.

Therefore, to make it a perfect cube we have to subtract 63.

792 – 63 = 729

729 is a perfect cube of 9.

### (i) 343 (ii) 2744 (iii) 4913 (iv) 1728 (v) 35937 (vi) 17576 (vii) 134217728 (viii) 48228544 (ix) 74088000 (x) 157464 (xi) 1157625 (xii) 33698267

Solution:

(i) 343

By prime factorizing 343, we get

∛343 = ∛ (7 × 7 × 7) = 7

Therefore, the cube root of 343 is 7

(ii) 2744

By prime factorizing 2744, we get

∛2744 = ∛ (2 × 2 × 2 × 7 × 7 × 7)

∛2744 = ∛ (23 × 73) = 2 × 7 = 14

Therefore, the cube root of 2744 is 14

(iii) 4913

By prime factorizing 4913, we get

∛4913 = ∛ (17 × 17 × 17) = 17

Therefore, the cube root of 4913 is 17

(iv) 1728

By prime factorizing 1728, we get

∛1728 = ∛(2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3)

∛1728 = ∛ (23 × 23 × 33) = 2 × 2 × 3 = 12

Therefore, the cube root of 1728 is 12

(v) 35937

By prime factorizing 35937, we get

∛35937 = ∛ (3 × 3 × 3 × 11 × 11 × 11)

∛35937 = ∛ (33 × 113) = 3 × 11 = 33

Therefore, the cube root of 35937 is 33

(vi) 17576

By prime factorizing 17576, we get

∛17576 = ∛ (2 × 2 × 2 × 13 × 13 × 13)

∛17576 = ∛ (23 × 133) = 2 × 13 = 26

Therefore, the cube root of 17576 is 26

(vii) 134217728

By prime factorizing 134217728, we get

∛134217728 = ∛ (227) = 29 = 512

Therefore, the cube root of 134217728 is 512

(viii) 48228544

By prime factorizing 48228544, we get

∛48228544 = ∛ (2 × 2 × 2 × 2 × 2 × 2 × 7 × 7 × 7 × 13 × 13 × 13)

∛48228544 = ∛ (23 × 23 × 73 × 133) = 2 × 2 × 7 × 13 = 364

Therefore, the cube root of 48228544 is 364

(ix) 74088000

By prime factorizing 74088000, we get

∛74088000 = ∛ (2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5 × 7 × 7 × 7)

∛74088000 = ∛ (23 × 23 × 33 × 53 × 73) = 2 × 2 × 3 × 5 × 7 = 420

Therefore, the cube root of 74088000 is 420

(x) 157464

By prime factorizing 157464, we get

∛157464 = ∛ (2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3)

∛157464 = ∛ (23 × 33 × 33 × 33) = 2 × 3 × 3 × 3 = 54

Therefore, the cube root of 157464 is 54

(xi) 1157625

By prime factorizing 1157625, we get

∛1157625 = ∛ (3 × 3 × 3 × 5 × 5 × 5 × 7 × 7 × 7)

∛1157625 = ∛ (33 × 53 × 73) = 3 × 5 × 7 = 105

Therefore, the cube root of1157625 is 105

(xii) 33698267

By prime factorizing 33698267, we get

∛33698267 = ∛ (17 × 17 × 17 × 19 × 19 × 19)

∛33698267 = ∛ (173 × 193) = 17 × 19 = 323

Therefore, the cube root of 33698267 is 323

### Question 5: Find the smallest number which when multiplied with 3600 will make the product a perfect cube. Further, find the cube root of the product.

Solution:

By prime factorizing 3600, we get

3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

By forming groups in triplet of equal factors we get, 3600 = (2 × 2 × 2) × (3 × 3) × (5 × 5) × 2

Since, 2, 3 and 5 cannot form a triplet of equal factors.

Therefore, 3600 must be multiplied with 60 (2 × 2 × 3 × 5) to get a perfect cube.

3600 × 60 = 216000

Cube root of 216000 is

∛216000 = ∛ (60 × 60 × 60)

∛216000 = ∛ (603) = 60

Therefore, the smallest number which when multiplied with 3600 makes a perfect cube is 60.

### Question 6: Multiply 210125 by the smallest number so that the product is a perfect cube. Also, find out the cube root of the product.

Solution:

By prime factorizing 210125, we get

210125 = 5 × 5 × 5 × 41 × 41

By forming groups in triplet of equal factors we get, 210125 = (5 × 5 × 5) × (41 × 41)

Since, 41 cannot form a triplet of equal factors.

Therefore, 210125 must be multiplied with 41 to get a perfect cube.

210125 × 41 = 8615125

Now, finding the cube root of 8615125

By using the prime factorization method, we get

8615125 = 5 × 5 × 5 × 41 × 41 × 41

Therefore, Cube root of product = ∛8615125 = ∛ (5 × 41) = 205

### Question 7: What is the smallest number by which 8192 must be divided so that quotient is a perfect cube? Also, find the cube root of the quotient so obtained.

Solution:

By prime factorizing 8192, we get

8192 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 23 × 23 × 23 × 2

By forming groups in triplet of equal factors we get, 8192 = (2×2×2)×(2×2×2)×(2×2×2)×(2×2×2)×2

Since, 2 cannot form a triplet of equal factors.

Therefore, 8192 must be divided by 2 to get a perfect cube.

8192/2 = 4096

Now, finding the cube root of 4096

By using the prime factorization method, we get

4096 = 2×2×2×2×2×2×2×2×2×2×2×2 = 23×23×23×23

Therefore, Cube root of 4096 = ∛4096 = ∛ (23×23×23×23) = 2×2×2×2 = 16

### Question 8: Three numbers are in the ratio 1:2:3. The sum of their cubes is 98784. Find the numbers.

Solution:

Given, ratio of number is 1:2:3

Therefore, Let the number be x, 2x and 3x

According to the question, sum of their cube is 98784

x3 + (2x)3+ (3x)3 = 98784

x3 + 8x3 + 27x3 = 98784

36x3 = 98784

x3 = 98784/36

x = 2744

x = ∛2744 = ∛ (2 × 2 × 2 × 7 × 7 × 7)

x = 2×7

x = 14

So, the respected numbers are,

x = 14

2x = 2 × 14 = 28

3x = 3 × 14 = 42

### Question 9: The volume of a cube is 9261000 m3. Find the side of the cube.

Solution:

Given, the volume of cube = 9261000 m3

Let the side of the cube be ‘x’ meter

Therefore, x3 = 9261000

Taking cube root on both the side,

x = ∛9261000 = ∛ (2×2×2×3×3×3×5×5×5×7×7×7) = ∛ (23×33×53×73) = 2×3×5×7 = 210

Hence, the side of cube is 210 meter

My Personal Notes arrow_drop_up