In this page we will explain the topics for the chapter 7 of Cube and Cube Roots Class 8 Maths.We have given quality notes and video to explain various things so that students can benefits from it and learn maths in a fun and easy manner, Hope you like them and do not forget to like , social share
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**Hardy - Ramanujan Numbers**

Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be

expressed as sum of two cubes in two different ways.

1729 = 1728 + 1 = 12

^{3} + 1

^{3}
1729 = 1000 + 729 = 10

^{3} + 9

^{3}
1729 is the smallest Hardy– Ramanujan Number. There are an infinitely many such numbers. Few are 4104 (2, 16; 9, 15), 13832 (18, 20; 2, 24), Check it with the numbers given in the brackets

**Cube Number**

Numbers obtained when a number is multiplied by itself three times are known as cube numbers

Example

1=1^{3}

8=2^{3}

27=3^{3}

Or if a natural number

*m *can be expressed as n

^{3} where

*n *is also a natural number, then

*m *is a

**cube number**
The numbers 1, 8, 27, 125 ... are cube numbers. These numbers are also called

**perfect ****cubes**.

**Some Important point to Note**

**Prime Factorization of Cubes**

When we perform the prime factorization of cubes number, we find one special property

8= 2×2×2 (Triplet of prime factor 2)

216 = (2 × 2 × 2) × (3 × 3 × 3) ( Triplet of 2 and 3)

Each prime factor of a number appears three times in the prime factorization of its cube.

**Smallest multiple that is a perfect cube**

Here we find the prime factorization of the number. Then we find the prime factor required to make all of them in triplet.

**Example**

Find the smallest multiple that will make 392 perfect cube

**Solution**:

392 = 2 × 2 × 2 × 7 × 7

The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect ube. To make its a cube, we need one more 7. In that case

392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744 which is a perfect cube

**Cube Root**

Cube root of a number is the number whose cube is given number

So we know that

27=3

^{3}
Cube root of 27

Cube root is denoted by expression

*How to Find Cube root*

**Finding cube root through prime factorization**

This method, we find the prime factorization of the number.

We will get same prime number occurring in pair for perfect square number. Square root will be given by multiplication of prime factor occurring in pair

Consider

1331

1331= (11×11×11)

**Example**
5832

**Solution**
5832= (2 × 2 × 2) × (3 × 3× 3) × (3 × 3× 3)

**Finding cube root by estimation method**

This can be well explained with the example

The given number is 17576.

**Step 1 **Form groups of three starting from the rightmost digit of 17576.

17 576. In this case one group i.e., 576 has three digits whereas 17 has only two digits.

**Step 2 **Take 576.

The digit 6 is at its one’s place.

We take the one’s place of the required cube root as 6.

**Step 3 **Take the other group, i.e., 17.

Cube of 2 is 8 and cube of 3 is 27. 17 lies between 8 and 27.

The smaller number among 2 and 3 is 2.

The one’s place of 2 is 2 itself. Take 2 as ten’s place of the cube root of

17576.

Thus,

**Extra Zing**

1) for any Positive integer m, m_{3} > m_{2} i.e cube is greater than square

2) For any negative integer m, m_{3} < m_{2} i.e cube is less than square , as the cube is always negative number and square is positive number

3) Cubes can be written as Addition consecutive odd numbers

1 = 1 = 1_{3}

3 + 5 = 8 = 2_{3}

7 + 9 + 11 = 27 = 3_{3}

13 + 15 + 17 + 19 = 64 = 4_{3}

21 + 23 + 25 + 27 + 29 = 125 = 5_{3}

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