Which of the following are not perfect cubes?
A.216
B.128
C.1000
D.100
E.46656

Hint: In order to proceed to find the perfect cube, write the prime factors of the number given. For a number to be a perfect cube every prime factor must appear 3 times or in multiples of 3.


Complete step-by-step answer:
216
The factors of 216 are ${2^3} \times {3^3}$
216 contain integers 2 and 3 which can be raised to the third power.
Hence, 216 is a perfect cube.

128
The factors of 128 are ${2^3} \times {2^3} \times 2$
128 contain integer 2 which cannot be raised to the third power due to the presence of one extra 2.
Hence, 218 is not a perfect cube.

1000
The factors of 216 are ${2^3} \times {5^3}$
1000 contain integers 2 and 5 which can be raised to the third power.
Hence, 1000 is a perfect cube.

100
The factors of 216 are ${2^2} \times {5^2}$
100 contains integers 2 and 5 which cannot be raised to the third power due to the lack of one 2 and one 5.
Hence, 100 is a perfect cube.

46656
The factors of 216 are ${2^3} \times {3^3} \times {2^3} \times {3^3} \times {2^3} \times {3^3}$
46656 contain integers 2 and 3 which can be raised to the third power.
Hence, 46656 is a perfect cube.

Hence, option (A), (C) and (E) are examples of perfect cube and the rest are not perfect cubes


Note: By doing the prime factorization of the number, the analysis of the number can be done completely.
The problems of perfect square, perfect cube, cube root and square root can be easily understood and solved using prime factorization.
For instance, $16$ whose factors are ${2^2} \times {2^2}$. Hence, it is a perfect square and its square root is a whole number i.e., 4.