Square Root Calculator

What is Square Root?

Square root is a basic mathematical operation that is used to solve the radical expression and find the root value of any positive number. The common input of square root is a positive number but its value is positive & negative. 

Both values are the same number, just differences in their direction that are shown by using the signs (+, -). i.e., the square root of 4 is +2 and -2. Here “+2” is called the principal root.

Components of Square Root

There are the three components of any square root expression, these are given below.

• Radical Sign/Symbol: It is shown as “√” that is used in any radical expression & also known as the square root symbol.  
• Radicand: Any number or expression that appears under the radical sign. 
• Index: The root/exponent value that represents the radical sign and shows the root type. i.e., square root, cube root, & fourth root.

Square Root Representation

The square root of any number is represented by the symbol “√” and written in a mathematical form for any variable say “x” such as: √x = y: where y × y = x

The above notation is known as the radical form due to the use of the square root symbol. The radical form can be defined in exponent or fraction form as:

                                   x^1/2 = y  ⇒  y² = x

If the index value is not mentioned, the square root value is conventionally taken to be equal to “1/2” as an exponent. To find the value in exponent form, use our exponent calculator.

How to Find Square Root?

There are four different manual methods to find the square roots of any positive number. But the best method is to use our square root calculator which calculates the root value instantly. However, manual methods to find square root are:

• Repeated Subtraction Method
• Prime Factorization Method
• Estimation (Approximation) Method
• Long Division Method

The first three methods are used to find the value of the perfect square number. But the fourth method is used for any number that is a perfect square or not. 

1. Calculating Square Root by Repeated Subtraction

In this method subtract the series of odd numbers (1, 3, 5, 7, 11, …) from a given number and continue until gets zero. The number of times subtractions are performed is the required square root value.

Note: This method only works for the perfect square numbers like (16, 81, 49, etc.). 

Example: Find the square root of “16” by repeated subtraction method.

Solution

Step 1: Subtract the series of odd numbers from 16 one by one.

16 - 1 = 15

15 - 3 = 12

12 - 5 = 7

   7- 7 = 0

Now, we note that the subtraction is performed 4-times. 

Thus, √16 = 4 by repeating subtraction method. Alternatively, use our above square root calculator that provides the answer quickly.

2. Find Square Root by Prime Factorization

In this method breaks a number into its prime factors by using a division table and makes pairs of them. Use prime factorization calculator to find prime factors of any number instantly. Also, see the below example to calculate the square root value with prime factorization method:

Example: Calculate the square root of “144” by factorization method.

Solution

Step 1: First, find the prime factor of “144” by factorization.

Prime factors of “144” = 2 × 2 × 2 × 2 × 3 × 3 

Step 2: Now, make the pairs of the same factor. 

Factors Pair = (2×2) × (2×2) × (3×3)

Step 3: Note the factor one time from each pair and multiply all of them.

Product of identical pair = 2 × 2 × 3 = 12

Thus, the square root of √144 is “12”. To verify this result use our simplified square root calculator and get the accurate answer in a single click.

3. Finding Root Value by Long Division

This trial division method is specially used to find the square root of those numbers that are too large, in decimal form, or not a perfect square. In this number divides by breaking into a sequence of easier steps and simply by division rules. For detailed steps, see the below example:

Example: Evaluate the square root of “19664” using long division.

Solution:

Step 1: First set the numbers in the square root symbol and make the pairs of numbers from right to left.

Step 2: Find a number whose multiple is less than or equal to the first pair/single digit.

 Here, the first number is 1. So, 1×1 = 1 ≤ 1. Now, divide the first number by “1”.

 Step 3: Continue this process by taking the next pair, until all pairs are used.

Thus, “140.228” is the square root value of “19664”. To verify the answer of the above imperfect square use our square root simplifier.

Some Well-Known Square Root Results

Sometimes we need to quickly add the result of a square root to solve any radical expressions or radical problems. For this, see the below table and remember some basic perfect square root values:

See the below table to remember some imperfect square root values:

FAQ’s about Square Root  

How to find square root of a number?

To find square root of a number manually then use one of the above-discussed methods. However, if you want to do this process instantly then use our square root calculator and calculate the sq. root of any positive number.

Can any number have more than one square root value?

Yes, all positive/non-negative numbers have a 2 square root value. One is positive and the other is negative but equal to the first.

What is the square root of 2?

The square root of 2 is an irrational number, approximately equal to “1.414213562”.

How to calculate the square root of a decimal number?

The long division method is commonly used to calculate the square root value of any decimal number. In this case, make the pairs of decimal & non-decimal parts separately and solve by following the rules of division.

Is square root calculator find square root accurately?

Yes, our simplify square root calculator is designed to give accurate results for both perfect & imperfect squares. You can use it easily on your computer or smartphone and calculate the square root of a given number.