Which of the following numbers are not perfect cubes?

(i) 216 (ii) 128


(iii) 1000 (iv) 100


(v) 46656


(i) Prime factorization of 216:

216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 × 33


Here we can see that each prime factor is appearing as many times as a perfect multiple of 3, hence, 216 is a perfect cube.


(ii)The prime factorization of 128 is:


128 = 2 × 2 × 2 × 2 × 2 × 2 × 2


Here, we can observe that 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 Hence, 128 is not a perfect cube.


(iii) The prime factorization of 1000 is:


1000 = 2 × 2 × 2 × 5 × 5 × 5


Here, we can observe that each prime factor is appearing as many times as a perfect multiple of 3.


Hence, 1000 is a perfect cube


(iv)The prime factorization of 100 is:


100 = 2 × 2 × 5 × 5


Here, we can see that every prime factor is not appearing as many times as a perfect multiple of 3.


Two 2s and two 5s are remaining if we group the triplets.


Therefore, 100 is not a perfect cube


(v)The prime factorization of 46656 is:


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


Here, we can see that number of 2s and 3s is 6 each and we know that 6 is divisible by 3.


Hence, 46656 is a perfect cube


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