 # Which of the following numbers are perfect cubes? Question:

Which of the following numbers are perfect cubes? In case of perfect cube, find the number whose cube is the given number.

(i) 125

(ii) 243

(iii) 343

(iv) 256

(v) 8000

(vi) 9261

(vii) 5324

(viii) 3375

Solution:

(i) 125

Resolving 125 into prime factors:

$125=5 \times 5 \times 5$

Here, one triplet is formed, which is $5^{3}$. Hence, 125 can be expressed as the product of the triplets of 5 .

Therefore, 125 is a perfect cube.

(ii) 243 is not a perfect cube.

(iii) 343

Resolving 125 into prime factors:

$343=7 \times 7 \times 7$

Here, one triplet is formed, which is $7^{3}$. Hence, 343 can be expressed as the product of the triplets of $7 .$

Therefore, 343 is a perfect cube.

(iv) 256 is not a perfect cube.

(v) 8000

Resolving 8000 into prime factors:

$8000=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5$

Here, three triplets are formed, which are $2^{3}, 2^{3}$ and $5^{3}$. Hence, 8000 can be expressed as the product of the triplets of 2,2 and 5, i.e. $2^{3} \times 2^{3} \times 5^{3}=20^{3}$.

Therefore, 8000 is a perfect cube.

(vi) 9261

Resolving 9261 into prime factors:

$9261=3 \times 3 \times 3 \times 7 \times 7 \times 7$

Here, two triplets are formed, which are $3^{3}$ and $7^{3}$. Hence, 9261 can be expressed as the product of the triplets of 3 and 7, i.e. $3^{3} \times 7^{3}=21^{3}$.

Therefore, 9261 is a perfect cube.

(vii) 5324 is not a perfect cube.

(viii) 3375 .

Resolving 3375 into prime factors:

$3375=3 \times 3 \times 3 \times 5 \times 5 \times 5$

Here, two triplets are formed, which are $3^{3}$ and $5^{3}$. Hence, 3375 can be expressed as the product of the triplets of 3 and 5, i.e. $3^{3} \times 5^{3}=15^{3}$

Therefore, 3375 is a perfect cube.