How To Find Square Root By Hand Advanced Methods (With Video) 2023

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Master the Art of Finding Square Roots: How To Find Square Root, Easy Methods and Practical Applications – Your comprehensive guide to square roots, from basics to advanced techniques. Unlock the world of numbers today!

Square roots can seem like a complex mathematical concept, but they need not be. Whether you’re a student struggling with math homework or someone simply curious about the world of numbers, this guide will help demystify the process of finding square roots. In this article, we will walk you through the basics, methods, and tricks for finding square roots.

What is a Square Root?

Before diving into the methods, let’s understand what a square root is. Simply put, the square root of a number is a value that, when multiplied by itself, equals the original number. For example, the square root of 25 is 5, because 5 * 5 equals 25.

Basic Method: Manual Calculation

The most fundamental way to find a square root is through manual calculation. This method is best suited for small numbers. Let’s break it down step by step:

  1. Estimate: Start by estimating the square root. For example, if you want to find the square root of 36, you can estimate it to be around 6 because 6 * 6 is 36.
  2. Guess and Check: Make a guess and square it. If the result is too high, try a lower number, and if it’s too low, try a higher number. In our example, start with 6 * 6 = 36. Since that’s correct, you’ve found the square root!

Read more: How to Find the Median: Formula, Examples, (2023)

Advanced Methods

For larger numbers or when you need a more precise result, there are advanced methods you can use.

1. Prime Factorization

This method is handy for finding square roots of large numbers. Here’s how it works:

  1. Prime Factorization: Decompose the number into its prime factors.
  2. Pair Them Up: Pair the prime factors in twos.
  3. Take One from Each Pair: Choose one factor from each pair.
  4. Multiply: Multiply all the chosen factors together.

Let’s apply this to find the square root of 144:

  • Prime factorization of 144: 2 * 2 * 2 * 2 * 3 * 3.
  • Pair them up: (2 * 2), (3 * 3).
  • Choose one from each pair: 2, 3,
  • Multiply: 2 * 3 = 6.

So, the square root of 144 is 6.

2. Long Division Method

The long division method is another precise way to find square roots, especially for numbers with multiple digits.

  1. Group the Digits: Group the digits of the number into pairs, starting from the decimal point and moving left. If there are an odd number of digits, add a zero to the left.
  2. Find the Largest Digit: Determine the largest number whose square is less than or equal to the leftmost group. This becomes the first digit of the square root.
  3. Subtract and Bring Down: Subtract the square of the first digit from the leftmost group, then bring down the next group of digits.
  4. Guess and Check: Append a decimal point to the result and continue with the process to find decimal places as needed.

3. Using a Calculator or Software

For quick and highly precise results, you can always rely on calculators or math software. Simply input the number, and the square root will be calculated instantly. This method is extremely convenient for complex calculations.

Practical Applications of Square Roots

Understanding square roots is not just a mathematical exercise; it has practical applications in various fields:

1. Engineering and Architecture

Engineers and architects often use square roots in calculations related to structural stability, load-bearing capacity, and material strength. Accurate square root calculations are crucial for ensuring the safety and integrity of buildings and bridges.

2. Finance and Economics

In finance, square roots are used to calculate volatility and risk in investment portfolios. Understanding square roots helps investors make informed decisions about their investments.

3. Computer Graphics

Square roots play a significant role in computer graphics, where they are used to calculate distances, angles, and scaling factors. This is essential for rendering realistic images in video games and simulations.

4. Statistics

Statisticians use square roots when calculating standard deviations, which measure the spread of data in a dataset. This is vital in fields such as epidemiology, economics, and the social sciences.


Finding square roots can be accomplished using various methods, ranging from basic estimation to advanced techniques like prime factorization and long division. Choose the method that suits your needs and the complexity of the number you’re working with.

Remember, practice makes perfect when it comes to finding square roots. The more you work with numbers, the more confident you’ll become in your calculations. So go ahead, unlock the mystery of square roots, and embrace the fascinating world of mathematics!

With this guide, you’ve taken the first step towards mastering the art of finding square roots. Whether you’re a student or simply curious about mathematics, understanding square roots will open doors to a deeper understanding of numbers and their relationships.

Now that you know how to find square roots, you can confidently tackle mathematical challenges with ease and precision. So, go ahead, practice, and become a square root expert!

Remember, the key to mastering any skill is practice, so keep exploring the world of numbers, and you’ll find that square roots are not as mysterious as they seem.

Happy calculating!


Q1: Can all numbers have square roots?

Yes, all real numbers have square roots, although the square roots of negative numbers are considered complex numbers.

Q2: What are imaginary square roots?

Imaginary square roots arise when you try to find the square root of a negative number. For example, the square root of -9 is represented as 3i, where ‘i’ stands for the imaginary unit.

Q3: How can I check if I found the correct square root?

To verify your answer, simply square the square root you found. If it equals the original number, you’ve got it right!

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