Question

How many perfect cubes lies between 1 and 1000

Answer 1

 we know that a perfect cube has a whole number as its cube root .

and the [tex] 10 ^{3} [/tex] = 1000 

so all numbers from 1 - 10 will have their cubes $\leq$ 1000

so therefore logically there exist 10 perfect cubes between 1 and 1000  

and the are :


1 - 1

2 - 8

3 - 27

4 - 64

5 - 125

6 - 216

7 - 343

8 - 512

9 - 729

10 - 1000

Answer 2

8 perfect cubes lie between 1 and 1000.

Perfect cubes between 1 and 1000 are as follows:

1. 2³ - 8

2. 3³ - 27

3. 4³ - 64

4. 5³ - 125

5. 6³ - 216

6. 7³ - 343

7. 8³ - 512

8. 9³ - 729

• In the prime factorization of a perfect cube, every prime factor occurs 3 times.

To determine whether a number is a perfect cube or not proceed as follows:

• Find the prime factors of the given number.
• Make a pair of 3 equal prime factors.
• If a group contains only one or two equal prime factors then a given number is not a perfect cube otherwise it is a perfect cube.

For e.g-

(1) 216 = (2 × 2 × 2 ) × ( 3 × 3 × 3)

216 has 2 groups of three equal prime factors.

Hence, 216 is a perfect cube.

(2) 128

128 = (2 × 2 × 2 ) x (2 × 2 × 2) × 2

Here, 128 has only 2 pairs of equal prime factors

Hence, 128 is not a perfect Cube.

(3) 100

100 = (2 × 2 ) × (5 × 5)

Here, the group (2 × 2 ) × (5 × 5) has only two prime factors.

Hence, 100 is not a perfect Cube.

Learn more on Brainly:

Which of the following is both a perfect square and a perfect cube? 16 64 125 144

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Question 1 Which of the following numbers are notperfect cubes? (i) 216 (ii) 128 (iii) 1000 (iv) 100 (v) 46656

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