Square Root Formula: Definition, Method and Solved Examples

Square root of a number is defined as the factor of a number which when multiplied by itself produces the original number as a result. The square root of any number can be obtained by multiplying the same number two times. The concept of square root is most essential in Mathematics as it has been used in our day to day life and in various mathematical calculations. The square of any number is a positive number. 

Read: Finding Square root

Table of Content

Key Terms: Number, Sum, Square, Positive Number, Multiplication, Factorisation,

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What is the Square Root Formula?

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The formula of square root helps in indicating the number in the form of its square root. The square root of any particular number is the value that we get by multiplying the same number with itself to give an original number. The square root of both a positive number and negative number will result in the same number. The square root can be represented by the symbol '√' in Mathematics. 

To understand it better let's take a simple example: The number four (4) has two square roots which are 2 and -2. This can be expressed as √4 = ±2. This can be achieved by (-2) × (-2) = 4 and 2 × 2 = 4.

Read More: Nature of Roots of Quadratic Equation

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Perfect Square Number and Square Root Formula

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Perfect number is defined as the number that is obtained by the product of an integer either by itself or as the second exponent of an integer number. For example, 25 is a perfect square number as it is a product of a number 5 by itself i.e. 5 × 5 = 25 whereas 21 is not a perfect square number as it cannot be obtained by the product of the number itself.

The square root of any particular number is the number raised by the power of ½. The results of the squares of both positive numbers and negative numbers are the same. However, the square root of any negative number can never be a real integer. The square root formula of any number x is represented by 

√x = x½. 

Now , suppose x is any number such that, x = p × p, then the formula for the calculation of square root of x is 

√x = √( p × p)

= √p²

= p

Here, we can say that p is the square root of x. That means if the value of the number p is an integer then x would be a perfect square number.

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Methods to Find Square Root

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There are different types of methods that can be used for calculating the square root. However, the long division method is the most common method for calculating the square root whether it is a square root or not. Some of the other methods for the calculation of square root are 

1. Prime Factorisation Method
2. Repeated Subtraction Method
3. Average Method
4. Long Division Method
5. Number Line Method

Prime Factorisation Method

In this method, the numbers are particularly expressed as the product of their prime factors. The prime factors which are identical in number are paired and the product of one number with each pair of the number results in the square root of a number. This method can be used in finding the square root of a number whether the number is a perfect square or not. But, the prime factorisation method cannot be used to find the square root of a decimal number that are not perfect squares. Below are given the steps to calculate the square root using the prime factorisation method.

• Step 1: Obtain the number given in question.
• Step 2: Now, solve the given number into prime factors through successive divisions.
• Step 3: Make pairs of each of the factors in such a way that each of the pairs are equal to each other.
• Step 4: Now take one factor from each of the pairs to find their products.
• Step 5: The product of the factors we get by multiplying the factor is the required square root.

To understand the above concepts more clearly let's find the square root of a number 576 using the prime factorisation method.

Repeated Subtraction Method

In this method, the particular number whose square root is to be calculated is repeatedly subtracted by the consecutive odd numbers till their differences become equal to the value of zero. The number of subtractions will give the root of that number. In this method, the square root of a number can be obtained by the perfect square numbers. 

For example, let us estimate the square root of 25. The number 25 is subtracted from odd numbers that start from the number 1.

Average Method of Square Root

The concept of the average method is used in this method for the calculation of the square root of a given decimal number. It is more conveniently used for the calculation of the square root of whole numbers upto a few decimal places. This concept can be more clear with the following example

For example, let us calculate the square root of 3 with the help of the average method. The number 3 lies between two square numbers that are 1 and 4. So, we can say that the square root of the number 3 lies in between 1 and 2. We can find the square root of 3 with the average of two square numbers.

By Long Division Method

The square of a number can be obtained by a long division method. The steps involved in finding the square of a number are given below:

• Step 1: First we pair the numbers from the unit place. If the number of digits is odd then we pair the leftmost digit of the number. 
• Step 2: Now take the biggest of the numbers as a divisor whose square number is less than or equal to the number given on the extreme left side.
• Step 3: Now bring the number one after another to the right side of the remainder.
• Step 4: Double the quotient value and enter on the right side of the number
• Step 5: Now guess the largest possible digit to which also becomes the new quotient. When the remainder becomes zero, the quotient becomes the square root of the number.

Finding Square of Odd Numbers as a Sum

The numbers which cannot be divided by the number 2 are called odd numbers. The square root of an odd number 'n' can be expressed as a sum of three consecutive positive integers.

For example, the sum of squares of 3 consecutive odd numbers can be obtained by 

= n (2n+1) (2n-1) / 3

= 3 (2×3+1) (2×3-1) / 3

= 35

1² + 3² + 5² = 1 + 9 + 25 = 35

Using Identity Property

The squares of any number having two or more digits can be found by writing the numbers as a sum of two different numbers. To make it more clear let's find the square of 21 using Identity Property.

21² = ( 20+1 )²

With the formula of (a+b)² = a² + b² + 2ab 

= 20² + 1² + 2 × 20 × 1

= 400 + 1 + 41

= 441

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Things to Remember

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• The square root of any square number is the number itself and the square of a square root of a number is the number itself.
• The number having 2,3, 7 or 8 at its units place is not a square number whereas the number having 0,1,4,5,6 or 9 at the units digit may or may not be a square number.
• The number whose units digit is either 1 or 9, then the square of this number has 1 at the unit place.
• The number having 6 at unit place, then if we square the number it will take 4 or 6 at the unit digit.
• Prime factorisation, average and repeated subtraction are the most common methods used for calculating the square root of any particular number.
• The Prime factorisation method can also be used for perfect squares whereas this method cannot be used in the calculation of square root of a decimal number.
• The concept of square root formula is widely used in a variety of applications in different fields of mathematics as well as physics.

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