Square Roots

What is Square Root ?

The square root of a natural number is a value, which can be written in the form of y = √a. It means ‘y’ is equal to the square root of a, where ‘a’ is any natural number. We can also express it as y2 = a. Thus, it is concluded here that square root is a value which when multiplied by itself gives the original number, i.e. a = y × y. Example: 4 × 4 =16, and the square root of 16 is 4.

Symbol of square root: The symbol or sign to represent a square root is ‘√’. This symbol is also called a radical. Also, the number under the root is called a radicand.

Square root examples:

  • 2 × 2 = 4 & √4 = 2
  • 3 × 3 = 9 & √9 = 3
  • 5 × 5 = 25 & √25 = 5
  • 6 × 6 = 36 & √36 = 6

In the above examples, it was easy to find the square root, since all the numbers are perfect squares here. But for imperfect squares, we do not get their square root as a natural number, instead, it will be a fraction.

Example: Square root of 2, √2 = 1.414

Finding Square Root of A Number By Prime Factorization

Square root of a number is the value that returns the original number on multiplied by itself. Finding square root by prime factorisation is an easy method. We need to factories the number under the root and pair them in two. For example, the square root of 9 is √9 = √(3×3) = 3.

How to Find Square Root By Prime Factorisation?

The inverse process of subtraction is addition and of division is multiplication. In the same way, the inverse of squaring a number is finding its root. For instance,

12 = 1, the square root of 1 is 1

42 = 16, square root of 16 is 4 and so on

Square Root by Prime Factorization Method

Prime factorization of any number means to represent that number as a product of prime numbers. To find the square root of a given number through the prime factorization method, we follow the steps given below:

  • Step 1: Divide the given number into its prime factors.
  • Step 2: Form pairs of factors such that both factors in each pair are equal.
  • Step 3: Take one factor from the pair.
  • Step 4: Find the product of the factors obtained by taking one factor from each pair.
  • Step 5: That product is the square root of the given number.

Find the square root of 324. 

The square root of 324 by prime factorization, we get


324 = 2 × 2 × 3 × 3 × 3 × 3

√324 = √(2 × 2 × 3 × 3 × 3 × 3)

= 2 × 3 × 3

SQUARE ROOT BY LONG DIVISION METHOD

In this section, you will learn, how to find square root of a number step by step.

Let us find the square root of 104976 step by step using long division method.

Step 1 :

Separate the digits by taking commas from right to left once in two digits.

10,49,76

When we do so, we get 10 before the first comma.

Step 2 :

Now we have to multiply a number by itself such that

the product ≤ 10

(The product must be greatest and also less than 10)

The above condition will be met by '3'.

Because 3 ⋅ 3 = 9 ≤ 10.

Now this situation is explained using long division.

In the above picture, 9 is subtracted from 10 and we got the remainder 1.

Step 3 :

Now, we have to bring down 49 and quotient 3 to be multiplied by 2 as given in the picture below.

Step 4 :

Now we have to take a same number at the two places indicated by '?'.

Then, we have to find the product as shown in the picture and also the product must meet the condition as indicated.

Step 5 :

The condition said in step 4 will be met by replacing '?' with '2'.

Than we have to do the calculation as given in the picture.

Step 6 :

Now, we have to bring down 76 and quotient 32 to be multiplied by 2 as given in the picture below.

 

Step 7 :

In the above picture, we have applied the procedures explained in step 4 and step 5. And we got the remainder zero.

Step 8 :

From the above picture, finally we got the square root of 104976. That is 324.

Hence, the square root of 104976 is 324