Cube Root of Numbers

Cube Root Introduction

In 1800 BCE, cube root was first used by Babylonian Mathematician who studied and practised Mathematics in Mesopotamia. Later, in the 4th century BCE, Plato, the Athenian Philosopher in ancient Greece came up with a problem of doubling the cube. Doubling the cube is a geometric problem where the given edge of the cube is doubled to get a cube of double volume. This is also known as the Delian problem.

We get to know the method of extracting cube roots from the 9th chapter of Chinese Mathematics book called Mathematical Art which was compiled around the 2nd Century BCE. In the first century CE the Greek mathematician Hero of Alexandria also devised a method for calculating cube roots. In 499 CE Aryabhata, an Indian mathematician and astronomer also gave a method for finding the cube root of numbers having many digits.

Cube of a Number

In order to find the cube of a number, we just have to multiply the number to itself three times. For example, cube of 4 is 4 x 4 x 4 which is equal to 64. The concept of cubing a number came from the volume of a cube. Thus, the cube of a 4 is nothing but the volume of a cube with side 4. Let us understand what a cube is.

What is a Cube?

There are geometric shapes all around us such as dice, square ice cubes, Rubik's cubes, square tissue boxes, sugar cubes, solid square tables and square pieces of cake, casserole, fudge or cornbread, all are geometric in shape. Cube is a 3-dimensional hexahedron with six faces such that three of them coincide at each vertex, and the angles between two faces are always 90 degrees. In simple words, a cube is made up of 6 squares of the same size. Every cube has six square faces, eight vertices and twelve edges.

The Volume of a Cube

The total internal space occupied by a cube is called the volume of a cube. The formula for the volume is the product of its length, breadth and height. Since, the length, breadth and height of a cube are equal to the side of the square that makes up the cube. Therefore. The volume of a cube is given by the given formula:

The volume of a cube = a x a  x a.

= a³.

The volume of a cube is a cube of the length of its side.

Perfect Cubes

If we calculate the volume of the cube above with volume (V), length (l), width (w) and height (h) as 5, then,

V = l x w x h

= 5 x 5 x 5

= 125

Now, 125 is the perfect cube because it is the product of an integer which is multiplied 3 times to itself. Hence, a perfect cube can be defined as a number that is the product of three equal integers. In simple words, a perfect cube is a number that can be achieved by cubing (multiplying 3 times to itself) a whole number.

Here are few more examples of perfect cube numbers:

1. Perfect cube of 2 = 2 x 2 x 2 = 8

2. Perfect cube of 3 = 3 x 3 x 3 = 27

3. Perfect cube of 4 = 4 x 4 x 4 = 64

4. Perfect cube of 5 = 5 x 5 x 5 = 125

5. Perfect cube of 6 = 6 x 6 x 6 = 216

6. Perfect cube of 7 = 7 x 7 x 7 = 343

7. Perfect cube of 8 = 8 x 8 x 8 = 512

8. Perfect cube of 9 = 9 x 9 x 9 = 729

9. Perfect cube of 10 = 10 x 10 x 10 = 1000

What is a Cube Root?

A cube root goes the opposite direction. For example, 3 is cubed to give the result 27 so the cube root of 27 will be 3.

Therefore, cube root of a number is a special number which when cubed gives the original number as a result. The cube root of 27 is 3 because 3 is cubed to produce 27.

Symbol of the Cube Root.

$\sqrt[3]{}$ is the symbol used to denote cube root. We can use it as $\sqrt[3]{27} = 3$ and we read it as “the cube root of 27 equals 3”

Cube Root of Negative Numbers

The cube of a negative number will also be a negative number.

-5 = -5 x -5 x -5 = -125

$\sqrt[3]{-125} = -5$

Properties of Cube Roots:

The table given below has the cubes of all the number between 11 to 20

From the table given above we can observe the following properties of cubes :

Property 1 :

A number having 1 in its unit digit, will also have 1 in the unit digit of their cubes.

For example,

1³  =  1

11³  =  1331

21³  =  9261

31³  =  29791

Property 2 :

The cubes of 1, 4, 5, 6, 9 and 0 also have the same digits in its unit digits.

For example,

14³  =  2744

15³  =  3375

16³  =  4096

20³  =  8000

Property 3 :

The cube of numbers ending 2 as unit digit will have 8 in its unit digit. Similarly, the cube of the numbers ending in unit digit 8 will have a unit digit 2.

For example,

12³  =  1728

18³  =  5832

Property 4 :

The cube of the numbers with 3 as unit digit will have a unit digit 7. Similarly, the cube of numbers with the unit digit 7 will have a unit digit 3.

For example,

13³  =  2197

27³  =  19683

Property 5 :

The cubes of all even numbers are even. Also, the cubes of odd numbers are all odd.

For example,

18³  =  5832 (even)

27³  =  19683 (odd)

Property 6 :

The sum of the cubes of the first natural numbers m is equal to the square of their sum.

That is,

1³ + 2³ + 3³ + ....... + m³  =  (1 + 2 + 3 + ..... + m)²

For example,

1³ + 2³ + 3³ + 4³  =  (1 + 2 + 3 + 4)²

1 + 8 + 27 + 64  =  (10)²

100  =  100

Solved Examples

Question 1: Find the cube root of the following:

• 27

• 125

Solution: Cube root of 27 is 3. (3 *3*3)

Cube root of 125 is 5 (5*5*5)

Q1. How to Find the Cube Root of an Integer?

Answer: The easiest way of how to find the cube root of a number is:

Make an estimate for the cube root. Never mind, if it is not close to the real root.

Example. To find the cube root of 756, divide the number by the estimate twice and take the average of estimate, estimate, and the remainder.

756/20 75.6

75.6/10=7.56

Average of 10,10,7.56 = 27.56/3 = 9.18.

This is not very close to the initial estimate.

So, repeat the above process taking the estimate as 9.

(Start with a round figure such that divisions can be done easily.)

756/9 = 84

84/9 = 9.33

Average of 9,9 and 9.33 = 27.33/3 = 9.11