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4.2

问题1.找到以下各项的多维数据集：
(i)-11
(ii)-12
(iii)-21

解决方案：

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

问题2.以下哪个数字是负整数的立方体。
(i)-64
(ii)-1056
(iii)-2197
(iv)-2744
(v)-42875

解决方案：

为了找出给定的负数是否是一个完美的立方体，我们需要检查其对应的正数是否是一个完美的立方体。

我)-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.

问题3。证明以下整数是负整数的立方体。另外，找到其立方为给定整数的整数：
(i)-5832(ii)-2744000

解决方案：

我)-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. 

问题4.找到以下项的多维数据集：
(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

解决方案：

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 = 3/10

So, 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 = 15/10

So, 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 = 8/100

So, 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 = 21/10

So, 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

问题5.以下哪个数字是有理数的立方：
(i)27/64
(ii)125/128
(iii)0.001331
(iv)0.04

解决方案：

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 = 5³/2³ × 2³ × 2

It 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) =11³/10³ × 10³

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