We have
64000 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5

Since, we get groups of triples.
\(\therefore\)64000 is a perfect cube.

We have, 3375
Resolving 3375 into prime factors, we get
3375 = 3 x 3 x 3 x 5 x 5 x 5
Clearly, the prime factors of 3375 are grouped into triples of equal factors and no factor is left over.
So, 3375 is a perfect cube.

We have, 53
One's digit of 53 = 3
Now, cube of one's digit of 53 = (3)³ = 3 x 3 x 3 = 27
Hence, the one's digit in the cube of 53 is 7.

We have, 1005
One's digit of 1005 = 5
Now, cube of one's digit of 1005 = (5)³ = 5 x 5 x 5 = 125
Hence,one's digit in the cube of 1005 is 5.

We have, 3331 = 1
One's digit of 3331 = 1
Now, cube of the one's digit of 3331 = (1)³= 1 x 1 x 1=1
Hence, one's digit in the cube of 3331 is 1.

\(\frac { 343 }{ 125 } \).

\(\frac { 2197 }{ 125 } \)

The prime factorisation of 5832
5832 = 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3
Therefore, \(\sqrt [ 3 ]{ 5832 } \)
= \(\sqrt [ 3 ]{ \underline { 2\times 2\times 2 } \underline { 3\times 3\times 3 } \underline { 3\times 3\times 3 } } \)
= 2 x 3 x 3
=18

Given, number is 614125.
So, groups of 614125 are

In the first group, the number 125 ends with 5. We know that, 5 comes at the unit's place of a number only when its cube ends in 5.
So, 5 will come at unit's place.
Now, in the second group, the number is 614
∵ (8)³ = 512 and (9)³=729
and 512 < 614 < 729
So, ten's place of required cube root is 8.
Hence, \(\sqrt [ 3 ]{ 614125 } \) = 85

Given, number is 21600.
Now, prime factorisation of 21600
21600 = 2 x 2 x 2 x 3 x 3 x 3 x 2 x 2 x 5 x 5
Prime factors 2 and 5 do not have triples.
So, 21600 is not a perfect cube.

14, 21, 28

Given, the volume of cube = 512 cm³
We know that, the volume of a cube = (Side)³
=(Side)³
Let the length of each side be a.
So, (a)³ = 512
a = \(\sqrt [ 3 ]{ 512 } \)
a = \(\sqrt [ 3 ]{ \underline { 2\times 2\times 2 } \times \underline { 2\times 2\times 2 } \times \underline { 2\times 2\times 2 } } \)
= 2 x 2 x 2
=8 cm
Hence, length of the each side of a cube is 8 cm.

8640

∵ 72 = 2 x 2 x 2 x 3 x 3
So, dividing by 3 x 3 = 9, we get a perfect cube.

∵ 250 = 5 x 5 x 5 x 2 ∴ Least number = 2 x 2 =4

(1.2)³ = (1.2) x (1.2) x (1.2) = 1.728

1 m =100 cm
So, 1m³ = 100x100 x 100 cm³ = 10⁶ cm³

8, [8, 27, 64, 125, 216, 343, 512, 729 are 8 perfect cubes between 1 and 1000]