Answer :

Let us understand, what perfect cube is.

A perfect cube is a number that is a cube of an integer.

So,

To find the perfect cube from the following numbers, we need to find their prime factors individually and then group them into triplets of the prime factors.

If no prime factor is left out, then the number is a perfect cube.

However, if one of the prime factors is a single factor or double factor, then the number is not a perfect cube.

Take 125.

Prime factors of 125 = 5 × 5 × 5

Group them into triplets.

⇒ Prime factors of 125 = (5 × 5 × 5)

Since, prime factor 5 form a triplet, so 125 is a perfect cube.

∴, 125 is a perfect cube.

Take 500.

Prime factors of 500 = 5 × 5 × 5 × 2 × 2

Group the factors into triplets.

⇒ Prime factors of 500 = (5 × 5 × 5) × 2 × 2

Since, prime factor 2 is not part of a triplet, so 500 is not a perfect cube.

∴, 500 is not a perfect cube.

Take 64.

Prime factors of 64 = 2 × 2 × 2 × 2 × 2 × 2

Group the factors into triplets.

⇒ Prime factors of 64 = (2 × 2 × 2) × (2 × 2 × 2)

Since, prime factor 2 forms two triplets, so 64 is a perfect cube.

∴, 64 is a perfect cube.

Take 7^{3}.

Prime factors of 7^{3} = 7 × 7 × 7

Group the factors into triplets.

⇒ Prime factors of 7^{3} = (7 × 7 × 7)

Since, prime factor 7 forms a triplets, so 7^{3} is a perfect cube.

∴, 7^{3} is a perfect cube.

Take 729.

Prime factors of 729 = 3 × 3 × 3 × 3 × 3 × 3

Group the factors into triplets.

⇒ Prime factors of 729 = (3 × 3 × 3) × (3 × 3 × 3)

Since, prime factor 3 forms two triplets, so 729 is a perfect cube.

∴, 729 is a perfect cube.

Take 968.

Prime factors of 968 = 2 × 2 × 2 × 11 × 11

Group the factors into triplets.

⇒ Prime factors of 968 = (2 × 2 × 2) × 11 × 11

Since, prime factor 11 is not a part of a triplet, so 968 is not a perfect cube.

∴, 968 is not a perfect cube.

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