Square Root
2026-02-28 18:04 Diff
https://brightchamps.com/en-us/math/algebra/square-root

There are different methods we use to calculate square roots of a given number. Some of the methods we use are given below:

Prime Factorization Method: 

While using the prime factorization method for square root calculation, a number is represented as the product of its prime factors. Remember that this method is only applicable to perfect square numbers. To find the square root of a perfect square, the prime factorization is the most commonly used method. We need to follow the steps mentioned below to calculate the square root of a number:

Step 1: Break down a number into its prime factors. Start with the smallest prime number, which is 2. 

Step 2: Group pairs of factors, where both factors in each pair are the same. 

Step 3: Take one factor from each pair. 

Step 4: Multiply the factors. The square root of a given number is the product of their factors.


For instance, take a look at this example of the square root of 144:

\(144 = 2 × 2 × 2 × 2 × 3 × 3\)

\((2 × 2) × (2 × 2)  × (3 × 3)\)

\((2 × 2 × 3)\)

\(144 = 122\)

\(√144 = 12\)

The square root of 144 is \( ±12\).

Long Division Method:
 

To find the square root of an imperfect number, we can apply the long division method. In this method, large numbers will be broken down into small parts.  The several steps of this method are:


Step 1: Break a number into pairs of two digits from right to left. 


Step 2: Find the greatest number whose square is smaller or equal to the first digit or pair.


Step 3: Now we can subtract the square of that number from the pair. After that, we can drop the next pair of numbers. 


Step 4: Double the number you found in step 2, and you get the new divisor.

 
Step 5: When we reach the required level, we can stop the steps. 

For example, we can find the square root of 20:

Step 1: We have one pair of digits for 20. 

Step 2: 4 is the largest number whose square is smaller than or equal to 20, because \((4 × 4 = 16)\). The first digit of the square root of 20 is ±4. 

Step 3: Subtract 16 from 20. 
\(20 - 16 = 4\)

Now, we have to drag down the next pair. But here, there is no other pair. So add a zero and make it 400. 

Step 4: Double the digit 4 and we get 8 as the result. 
 

Step 5: Next, we find the largest number, whose square is less than or equal to 400, when we multiply by the new number and add to 8. Here, 5 is the next digit. 
\(8 × 5 = 40\)
\(40 × 5 = 200\)
 

Step 6: Subtract 200 from 400 which gives 200. 
Now, we know the approximate square root of 20 is ±4.5. 

Using a Calculator:
 

This is a simple and interesting way to calculate the square roots of any given number. Finding the symbol of square root √ in the calculator is the foremost thing you should remember. Just enter your number, then press the square root symbol √. The calculator will display the square root.