第8类RD Sharma解决方案-第4章多维数据集和多维数据集根–练习4.3

问题1：通过依次减去数字，找到以下数字的立方根：1、7、19、37、61、91、127、169、217、271、331、397，…

(i)64

(ii)512

(iii)1728年

解决方案：

(i) 64

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.

问题2：使用逐次减法的方法检查以下数字是否为理想的立方体：

(i)130

(ii)345

(iii)792

(iv)1331年

解决方案：

(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.

问题3：找出必须从问题2中不是理想立方体的数字中减去的最小数，以使它们成为理想立方体。对应的多维数据集根是什么?

解决方案：

在上一个问题中，有三个数字不是完美的立方体。

(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.

问题4：找到以下每个自然数的立方根：

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

解决方案：

(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 

问题5：找出最小的数字，将其乘以3600将使乘积成为一个完美的立方体。此外，找到产品的立方根。

解决方案：

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.

问题6：将210125乘以最小数，以使乘积是一个理想的立方体。此外，找出产品的立方根。

解决方案：

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

问题7： 8192必须除以的最小数是多少，才能使商成为一个完美的立方?同样，找到这样获得的商的立方根。

解决方案：

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

问题8：三个数字的比例为1：2：3。他们的立方体的总和是98784。找到数字。

解决方案：

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

问题9：立方体的体积为9261000 m ³ 。找到立方体的一面。

解决方案：

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