Squares, Cubes and Roots

If a given number is multiplied by itself, we say that the result is the square of the number. Similarly if the number is multiplied by itself three times, the result is said to be the cube of the number.  Reverse of squares and cubes are called square roots or cube roots of the numbers respectively.

What we will study in this chapter :

1. Square and cube of a number
2. Square roots and cube roots of a number
3. Where are they useful?

1. Square and cube of a number
For example, if  5 is multiplied by itself, the result is 25.

5 * 5 = 25. 

The number 25 is the square of 5. Conventionally, it is said that 5 squared is 25. 

If 5 is multiplied thrice by itself, the result is 125. 

5 * 5 * 5 = 125 

The number 125 is the cube of 5 or 5 cubed is 125. 

There are many ways to write squares and cubes. 
One way is as follows : 5² = 25, 5³ = 125. The numbers raised are called indices. We will study a bit more about indices in later chapters. We say that 5 raised to 2 is 25, or 5 raised to 3 is 125.

It is easy to determine the squares and cubes of numbers, as one has to multiply the same number either twice or thrice respectively.

2. Square roots and cube roots of a number
The reverse of squares or cubes is called square roots or cube roots. For example, the square root of 25 is 5, or the cube root of 125 is 5. 

5 squared is 25, square root of 25 is 5. The square root is written as 25 = 5

5 cubed is 125, cube root of 125 is 5. The cube root is written as   ³125        = 5

There are other ways to write square roots and cube roots. In terms of indices they can be written as 

25 ^(1/2) = 5. 25 raised to half is 5, which is same as the square root of 25.

125 ^(1/3) = 5. 125 raised to one third is 5, which is the same as the cube root of 125.

It has to be remembered that square of (-5) is (-5) * (-5) = 25, the square root of 25 thus can be either 5 or ñ5.

Thus square root of a number can either have a positive or a negative result. 

Cube root of a positive number will always be a positive number. Cube root of a negative number will always be a negative number.

To find the square roots or cube roots of a number, some arithmetic needs to be done.  Multiplication tables have to be known by heart. Finding square roots or cube roots of a number by factorization is relatively a simple procedure. But not all numbers can be factorized neatly. In that case other methods like finding square roots by division method, finding cube roots by table method, etc. have to be followed. 

As we advance into the chapter of indices and logarithmic tables, methods for finding square and cube roots will become easier.

3. Where are they useful? 
Squares and cubes and their roots are extensively used in determining areas, volumes of surfaces. For example, area of a square whose one side is r, is r² or r squared.

Area = r * r =r2

If you know the value of area, you can find the length of the side by using the square root of the area. Thus area   = r. ( the second root ñr is physically meaningless) 

Volume = r * r * r =r3

If you know the volume of the cube, you will be able to determine the length of the side of the cube by using the cube root of the volume.

Thus  ³volume  = r

The examples given above are the simplest ones. As we learn advanced geometry of surfaces and volumes, we will get to know where the squares, cubes and their roots are useful.

Next        Main        Previous