Class 8 Maths Cubes and Cube Roots | Cubes |

__Cubes__

A figure that has 3 dimensions are called solid figures.

A cube is a solid figure which has all its sides equal i.e. its length, width and height are equal.

Let us consider a cube having dimension 2 cm i.e. its

Length, width and height are 2 cm each.

We can calculate volume of a cube = length * width * height

= 2 * 2 * 2

= 8 cm^{3}

Now, consider the numbers 1, 8, 27, etc.

1 = 1 * 1 * 1 = 1^{3}

8 = 2 * 2 * 2 = 2^{3}

27 = 3 * 3 * 3 = 3^{3}

Each of these numbers is obtained when a number is multiplied by itself three times. These numbers are called perfect cubes or cube numbers.

The following are the cubes of numbers from 1 to 10.

From the above table, we can observe that cube of an even number is even and cube of an odd number is odd.

**Some interesting patterns**

1. **Adding consecutive odd numbers**

** **Observe the following pattern of sums of 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}

and so on.

**Cubes and their prime factors**

Consider the following prime factorization of the numbers and their cubes.

** ****Prime factorization of a number Prime factorization of its cube**

** **4 = 2 * 2 4^{3} = 64 = 2 * 2 * 2 * 2 * 2 * 2 = 2^{3} * 2^{3}

6 = 2 * 3 6^{3} = 216 = 2 * 2 * 2 * 3 * 3 * 3 = 2^{3} * 3^{3}

15 = 3 * 5 15^{3} = 3375 = 3 * 3 * 3 * 5 * 5 * 5 = 3^{3} * 5^{3}

12 = 2 * 2 * 3 12^{3} = 1728 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3

= 2^{3} * 2^{3} * 3^{3}

Now, we observe that each prime factor of a number appears three times in the prime factorization of its cube. So, in the prime factorization of any number, if each factor appears three times then the number is a perfect cube.

Let us take some examples.

Let we have two numbers 216 and 750

By prime factorization, 216 = 2 * 2 * 2 * 3 * 3 * 3

Here, each factor appears 3 times. 216 = 2^{3} * 3^{3} = (2 * 3)^{3} = 6^{3} which is a perfect cube.

Again 750 = 2 * 3 * 5 * 5 * 5 = 2 * 3 * 5^{3}

Here, each factor does not appear 3 times. So, it is not a perfect cube.

**Smallest multiple that is a perfect cube **

There are many numbers which are not a perfect cube. But we can make these numbers as a perfect cube by multiplying or dividing the smallest number.

Let us take some number 36 and 250.

By prime factorization, 36 = 2 * 2 * 3 * 3

The prime factor 2 and 3 do not appear in a group of three. Therefore, 36 is not a perfect cube. To make it a cube, we need one more 2 and 3. In that case

36 * 2 * 3 = 2 * 2 * 2 * 3 * 3 * 3 = 216, which is a perfect cube.

Hence, the smallest natural number by which 36 should be multiplied to make a perfect cube is 6.

Again by prime factorization, 250 = 2 * 5 * 5 * 5

The prime factor 2 does not appear in a group of three. Therefore, 250 is not a perfect cube. To make it a cube, we divide the number by 2 so that the prime factorization of the quotient will not contain 2.

In that case 250/2 = 5 * 5 * 5 = 125, which is a perfect cube.

Hence, the smallest natural number by which 250 should be divided to make a perfect cube is 2.

**Problem: Ramesh makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will ****he need to form a cube?**

**Solution:**

Volume of the cube of sides 5 cm, 2 cm 5 cm = (5 * 2 * 5) cm^{3}

Here, two 5s and one 2 are left which are not in a triplet.

If we multiply this expression by 2 * 2 * 5 = 20, then it will be a perfect square.

Thus, 5 * 2 * 5 * 2 * 2 * 5 = 2 * 2 * 2 * 5 * 5 * 5 = 1000

This is a perfect square.

Hence, 20 cuboids of 5 cm, 2 cm, 5 cm are required to form a cube.

.