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Which of the following numbers are notperfect cubes?

(i) 216

(ii) 128

(iii) 1000

(iv) 100

(v) 46656

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(i) The prime factorisation of 216 is as follows. 2 216 2 108 2 54 3 27 3 9 3 3 1 216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 × 33 Here, as each prime factor is appearing as many times as a perfect multiple of 3, therefore,...
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(i) The prime factorisation of 216 is as follows.

2

216

2

108

2

54

3

27

3

9

3

3

 

1

216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 × 33

Here, as each prime factor is appearing as many times as a perfect multiple of 3, therefore, 216 is a perfect cube.

(ii)The prime factorisation of 128 is as follows.

2

128

2

64

2

32

2

16

2

8

2

4

2

2

 

1

128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

Here, each prime factor is not appearing as many times as a perfect multiple of 3. One 2 is remaining after grouping the triplets of 2. Therefore, 128 is not a perfect cube.

(iii) The prime factorisation of 1000 is as follows.

2

1000

2

500

2

250

5

125

5

25

5

5

 

1

1000 = 2 × 2 × 2 × 5 × 5 × 5

Here, as each prime factor is appearing as many times as a perfect multiple of 3, therefore, 1000 is a perfect cube.

(iv)The prime factorisation of 100 is as follows.

2

100

2

50

5

25

5

5

 

1

100 = 2 × 2 × 5 × 5

Here, each prime factor is not appearing as many times as a perfect multiple of 3. Two 2s and two 5s are remaining after grouping the triplets. Therefore, 100 is not a perfect cube.

(v)The prime factorisation of 46656 is as follows.

2

46656

2

23328

2

11664

2

5832

2

2916

2

1458

3

729

3

243

3

81

3

27

3

9

3

3

 

1

46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

Here, as each prime factor is appearing as many times as a perfect multiple of 3, therefore, 46656 is a perfect cube.

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