# Square Root of a Number(Meaning, Formulas & Examples)

This post is also available in: हिन्दी (Hindi)

You know the area of a rectilinear figure square = $\left(\text{side} \right) \times \left(\text{side} \right)$, where $\text{side}$ is the length of edge(or side) of a square. If you want to find the length of an edge of a square whose area is known. How will you do that? In such cases, we find the square root of a number. For example, if area of a square is $256 \text{cm}^2$, then the length of its edge(or side) is $16 \text{cm}$.

Let’s understand what is a square root of a number, how to find the square root of a number, and what are the properties of the square root of a number.

## What is Square Root?

The square root of a number is that factor of a number which when multiplied by itself gives the original number. Squares and square roots are special exponents.

For example, consider the number $25$. When $5$ is multiplied by itself, it gives $25$ as the product. This can be written as $5 \times 5$ or $5^{2}$. Here, the exponent is $2$, and we call it a square. Now when the exponent is $\frac{1}{2}$, it is called as the square root of the number. For example, $\sqrt{n}=n^\frac{1}{2}$, where $n$ is a positive integer.

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

If square of a number $a$ is $b$, then square root of the number $b$ is $a$.

## How to Find the Square Root of a Number?

It is very easy to find the square root of a number that is a perfect square. Perfect squares are those positive numbers that

can be expressed as the product of a number by itself. in other words, perfect squares are numbers that are expressed as the value of power $2$ of any integer. For example, $9$ is a perfect square number, since $3^{2} = 9$. We can use any of these four methods to find the square root of numbers.

• Repeated Subtraction Method of Square Root
• Square Root by Prime Factorization Method
• Square Root by Estimation Method

### Repeated Subtraction Method of Square Root

This is a very simple method. In this case, we use one of the properties of square numbers. The property states that the “Sum of first $n$ odd numbers is equal to $n^{2}$”

We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of $36$ using this method.

$36 – 1 = 35$

$35 – 3 = 32$

$32 – 5 = 27$

$27 – 7 = 20$

$20 – 9 = 11$

$11 – 11 = 0$

You can observe that we have subtracted $6$ times. Thus, $\sqrt{36} = 6$.

### Square Root by Prime Factorization Method

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 similar 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.

Let’s consider some examples to understand the process.

### Examples

Let’s consider some examples to understand the process.

Ex 1: Find the square root of $144$.

Prime factorization of $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^{4} \times 3^{2} = \left(2^{2} \right)^{2} \times 3^{2} = 4^{2} \times 3^{2} = (4 \times 3)^{2} = 12^{2}$.

### What are the applications of the square root formula?

There are various applications of the square root formula:
a) The square root formula is mainly used in algebra and geometry.
b) It helps in finding the roots of a quadratic equation.
c) It is widely used by engineers.