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 cm3

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

1 = 1 * 1 * 1 = 13

8 = 2 * 2 * 2 = 23

27 = 3 * 3 * 3 = 33

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

Observe the following pattern of sums of odd numbers.

1 = 1 = 13

3 + 5 = 8 = 23

7 + 9 + 11 = 27 = 33

13 + 15 + 17 + 19 = 64 = 43

21 + 23 + 25 + 27 + 29 = 125 = 53

and so on.

1. 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                                       43 = 64 = 2 * 2 * 2 * 2 * 2 * 2 = 23 * 23

6 = 2 * 3                                         63 = 216 = 2 * 2 * 2 * 3 * 3 * 3 = 23 * 33

15 = 3 * 5                                     153 = 3375 = 3 * 3 * 3 * 5 * 5 * 5 = 33 * 53

12 = 2 * 2 * 3                          123 = 1728 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3

=  23 * 23 * 33

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 = 23 * 33 = (2 * 3)3 = 63 which is a perfect cube.

Again 750 = 2 * 3 * 5 * 5 * 5 = 2 * 3 * 53

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) cm3

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.

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