The concept of square roots is very important in mathematics. It appears in various calculations, from algebra to physics and engineering. Understanding the concept of square root and how to find the square root of a number is needed for solving complex mathematical problems.
If you have ever wondered about how to calculate square root of any number, we will provide a detailed guide below to help you master the concept.
A square root of any given number is a unique value that gives the original number when it is multiplied by itself. The symbol for the square root is √, which is called the radical symbol. The number under the radical symbol is called the radicand. Mathematically, if x² = y, then x is the square root of y. For example, the square root of, 81 is 9 because 9 x 9 gives 81.
To understand how you find a square root, you need to go through the basics of the concept. For how do you find out the square root, there are several methods, each suited for different scenarios. Let’s explore some of the best tried and tested methods for calculating the square root of a number.
It is a method to find the square root of a number by breaking it down into the prime factors. At first, you need to break the numbers into their prime factors. Then, you need to group the prime factors into pairs. You need to choose one factor from each pair and multiply the chosen factors together. The end result is the square root of the original number. This method is best suited for perfect squares.
Example: Find the square root of 144.
Prime factorisation of 144 is: 2 x 2 x 2 x 2 x 3 x 3
Grouping: (2 x 2), (2 x 2), (3 x 3)
Taking one number from each pair: 2 x 2 x 3 = 12
So, √144 = 12
The long-division method in mathematics is a method for dividing large numbers by breaking the problem into smaller steps. It is a popular method for how to calculate square root of any number.
To apply this method, you need to group the digits of the number in pairs from right to left. Then, you need to find the largest number whose square is less than or equal to the first pair. You should then subtract and bring down the next pair of digits. After that, you should double the quotient and find the next divisor. Keep repeating the process until the desired accuracy is achieved.
Example: Find the square root of 50 using long division.
Here is a step-by-step guide on how to find the square root of 50 using long division.
For quick calculations, estimation and approximation are very useful. To apply this method, you need to find the nearest perfect square of the number. Then, one needs to determine which square root value the number is closest to. You can interpolate between the values if needed.
Example: Find the value of √72.
The nearest perfect squares are 64, which is 8² and 81, which is 9².
Since 72 is closer to 64, we estimate √72 ≈ 8.5.
For complex numbers, the formula for the square root of a number is:
√N ≈ Initial Guess + (N - extInitialGuess²/ 2 x extInitialGuess).
Example: Find √20 using 4.5 as the initial guess.
√20 ≈ 4.5 (20 - 4.5²/ 2 x 4.5) = 4.5 - 0.0278 ≈ 4.47
If you do not have a calculator and you are wondering how to calculate square root of any number, utilise the following for an accurate result:
Let’s explore some square roots of a number of examples for a better understanding of the concepts.
Here are some “how to find square root practice questions” to strengthen your understanding.
Understanding how to find the square root of a number can help in various mathematical and real-world applications. Using the formula for the square root of a number will make students more proficient in solving complex mathematical equations. Keep practising diligently to improve your skills.
(Session 2025 - 26)