Square roots are essential in mathematics, especially when dealing with geometry, algebra and real-life applications such as measurements, architecture, and many others. Despite the calculators’ convenience, knowing how to calculate square roots by hand is a valuable skill to acquire and maintain. In this article, we will walk you through the process of finding the square root of a number by hand using two simple methods – the Prime Factorization method and the Long Division method.

Method 1: Prime Factorization

Step 1: Write down the number

Identify the number whose square root you want to find and write it down.

Step 2: Prime factorize the number

Divide the given number into its smallest prime factors that multiply together to produce that number. You can do this by dividing the number with prime numbers (2, 3, 5, 7…) until you cannot divide it any further.

Step 3: Group the factors into pairs

Arrange the obtained prime factors in ascending order and group them into pairs. Each pair should have two equal factors.

Step 4: Write down one factor from each pair

Choose one factor from each pair and write them down in sequence.

Step 5: Multiply those factors

Multiply all those factors you wrote in Step 4 together. The result is your square root.

Method 2: Long Division

Step 1: Write down the number

Write down the number whose square root you want to find as a four-digit number (you can include trailing zeroes if needed).

Step 2: Pair up digits

Group the digits of your four-digit number into pairs starting from right to left. If there is an odd number of digits, treat the left ungrouped digit as a separate group.

Step 3: Divide and subtract

Find a perfect square closest to your first group on the left. Subtract that closest perfect square from the first group, and write the square root of it as the divisor.

Step 4: Bring down and double

Bring down the next group and place it to the right of your subtraction’s remainder. Double your current divisor, ignore the remainder, and write it on a new line next to the dividend.

Step 5: Create updated divisor and quotient

Choose a digit to append to your doubled divisor so that when you multiply it by itself (the digit), the product is equal to or less than the updated dividend. Write this digit beside your doubled divisor, creating an updated divisor, and write the same digit in your quotient (above the dividend).

Step 6: Repeat Steps 3-5

Keep repeating Steps 3-5 for any remaining groups of numbers until you find your square root. The digits appear sequentially in your quotient are the approximate square root.

Conclusion

Calculating a square root by hand is a great mental exercise that not only helps you gain better number sense but also trains your brain’s problem-solving abilities. While both methods mentioned above have their advantages, use whichever method works best for you or even try other methods such as Newton-Raphson iteration or Babylonian method to find an approach that suits you best.