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Which of the following numbers are notperfect cubes?

(i) 216 (ii) 128 (ii) 1000

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(i) The prime factorisation of 216 is as follows.

2216
2108
254
327
39
33
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.

2128
264
216
28
24
22
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

21000
2500
2250
5125
525
55
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.

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