(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 |
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.