Q. 44.5( 13 Votes )

# 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) 324 (viii) 3375

Answer :

(i) 125

First find out the prime factors of 125,

125 = 5×5×5

As we see a group of three 5 is made, which we can also be write as 5^{3};

So, 125 is the product of triplets of 5.

Therefore, it is the perfect cube.

(ii) 243

The prime factorization of 256 is shown below:

243 = 3 × 3 × 3 × 3 × 3

To be a perfect cube the prime factors of number should make a group of 3 but as we can see here more than 3 numbers are available in prime factors.

So, 243 is not the perfect cube.

(iii) 343

The prime factorization of 256 is shown below:

343 = 7 × 7 × 7

As we see a group of three 7 is formed, which we can also be write as 7^{3};

So, 343 is the product of triplets of 7.

Therefore, it is the perfect cube.

(iv) 256

The prime factorization of 256 is shown below:

256 = 2×2×2×2×2×2×2×2

If the prime factors are not making the pairs of three so the number is not perfect cube.

(v) 8000

The prime factorization of 8000 is shown below:

8000 = 2×2×2×2×2×2×5×5×5

As we can see three pairs can be made of the above prime factors, which are 2^{3}, 2^{3}, and 5^{3}.

So, 8000 can be expressed as the product of the triplets of 2, 2 and 5, i.e.

2^{3} × 2^{3} × 5^{3} = 20^{3}

Therefore, 8000 is a perfect cube.

(vi) 9261

The prime factorization of 9261 is shown below:

9261 = 3×3×3×7×7×7

As we can see two pairs can be made of the above prime factors, which are 3^{3}, and 7^{3}.

So, 9261 can be expressed as the product of the triplets of 3 and 7, i.e.

3^{3} × 7^{3} = 21^{3}

Therefore, 9261 is a perfect cube.

(vii) 5324

The prime factorization of 5324 is shown below:

5324 = 2×2×11×11×11

Therefore, 5324 is not a perfect cube.

(viii) 3375

The prime factorization of 3375 is shown below:

3375 = 3×3×3×5×5×5

As we can see two pairs can be made of the above prime factors, which are 3^{3}, and 5^{3}.

So, 3375 can be expressed as the product of the triplets of 3 and 5, i.e.

3^{3} × 5^{3} = 15^{3}

Therefore, 3375 is a perfect cube.

Rate this question :

Which of the following numbers are not perfect cubes?

(i) 64

(ii) 216

(iii) 243

(iv) 1728

RD Sharma - MathematicsFind the cubes of the following numbers by column method:

(i) 35

(ii) 56

(iii) 72

RD Sharma - MathematicsWrite the units digit of the cube of each of the following numbers:

31, 109, 388, 4276, 5922, 77774, 44447, 125125125

RD Sharma - MathematicsWhich of the following are cubes of odd natural numbers?

125, 343, 1728, 4096, 32768, 6859

RD Sharma - MathematicsFind the volume of a cube whose surface area is 384m^{2}.

Choose the correct answer:

Write the cubes 5 natural numbers of which are multiples of 7 and verity the following:

“The cube of a multiple of 7 is a multiple of 7^{3}.

By what least number should 324 be multiplied to get a perfect cube?

RS Aggarwal - Mathematics