A square root is a value that gives the original number that multiplication of itself. e.g., 6 multiplied by itself gives 36 (i.e., 6 × 6 = 36), therefore, 6 is the square root of 36, or in other words, 36 is the square number of 6.
Common methods to find square roots-
1. Repeated Subtraction Method
The sum of the first n odd natural numbers is known to be n2. Using this idea, we repeatedly subtract consecutive odd numbers starting from 1 until we get 0.
The number of steps required gives the square root.
Example: Determine the square root of 25 using the repeated subtraction method.
Solution:
- 25 - 1 = 24
- 24 - 3 = 21
- 21 - 5 = 16
- 16- 7 = 9
- 9 - 9 = 0
Here it takes five steps to get the 0. Hence, the square root of 25 is 5.
This method works best for small perfect squares.
2. Prime Factorization Method
In this method, the number is written as a product of its prime factors. Pairs of equal prime factors are formed, and one factor from each pair is multiplied to find the square root.
e.g.: The prime factors of 126 will be 2, 3 and 7 as 2 × 3 × 3 × 7 = 126 and 2, 3, 7 are prime numbers.
- 16 = 2 × 2 × 2 × 2 = 22 × 22 = √16 = 4
- 25 = 5 × 5 = 52 = √25 = 5
- 64 = 2 × 2 × 2 × 2 × 2 × 2 = √64 = 8
This method cannot be used for non-perfect squares or decimal numbers.
3. Division Method
For large numbers, prime factorization becomes lengthy. The long division method is a systematic way to find square roots of perfect squares and decimals.
Steps:
- Group the digits of the number in pairs from the right.
- Find the largest number whose square is less than or equal to the leftmost group.
- Use division and subtraction.
- Bring down the next pair and repeat the process.
- Continue until the remainder becomes zero.
Example : Find the square root of 144
Solution:
The steps to determine the square root of 144 are:
Step 1: Start the division from the leftmost side. Here 1 is the number whose square is 1.
Step 2: Putting it in the divisor and the quotient and then doubling it will give as,
Step 3: Now it is required to find a number for the blanks in divisor and quotient. Let that number be x.
Step 4: Therefore, check when 2x multiplies by x give a number of less than or equal to 44. Take x = 1, 2, 3, and so on and check.
In this case,
- 21 × 1 = 21
- 22 × 2 = 44
So we choose x = 2 as the new digit to be put in the divisor and in the quotient.
The remainder here is 0 and hence 12 is the square root of 144.
4. Square Root by Estimation Method
This method is used when the number is not a perfect square. We estimate the square root using nearby perfect squares.
Example: Estimate √65
Identify the perfect squares closest to 65. The perfect square smaller than 65 is 64 (82) and the perfect square larger than 65 is 81 or 92.
Since 65 is closer to 64, start with the square root of 64, which is 8.
Determine whether you need to increase or decrease your estimate based on how far 65 is from the perfect squares.
- 82 = 64, which is less than 65.
- 92 = 81, which is more than 65.
Since 65 is closer to 81, increase the estimate.
Take the average of 8 and 9: (8+9)/2 = 8.5
Test (8.52): 8.5 × 8.5 = 72.25
Since 72.25 is less than 65, further increase the estimate.
- Another average: (8.5+9)/2 = 8.75
- Test (8.752): 8.75 × 8.75 = 76.5625
Since 76.5625 is greater than 65, stick with the previous estimate.
The estimated square root of 65 is approximately 8.75
This estimation method provides a reasonable approximation of the square root, especially when the number is not a perfect square.