Which of the following numbers are not perfect cubes? (i) 216 (ii) 128 (iii) 1000 (iv) 100 (v) 46656

(i) We have 216 = 2 x 2 x 2 x 3 x 3 x 3

Grouping the prime factors of 216 into triples, no factor is left over.
∴ 216 is a perfect cube.
(ii) We have 128 = 2 x 2 x 2 x 2 x 2 x 2

∴ 128 is not a perfect cube.
(iii) We have 1000 = 2 x 2 x 2 x 5 x 5 x 5

∴ 1000 is a perfect cube.
(iv) We have 100 = 2 x 2 x 5 x 5

Grouping the prime factors into triples, we do not get any triples.
Factors 2 x 2 and 5 x 5 are not in triples.
∴ 100 is not a perfect cube.
(v) We have 46656 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3

Grouping the prime factors of 46656 in triples we are not left over with any prime factor.
∴ 46656 is a perfect cube.