When we square a number, we multiply the number by itself. Square roots are the exact opposites: we divide the square by the number that was multiplied by itself. Square roots are denoted by the symbol √. In this blog, I will talk about how to find square roots with just a pen and a piece of paper, or even mentally.

Finding square roots using prime factorization

When we prime factorize, we keep dividing a number with prime numbers until the quotient is 1. We can use it to find square roots. Let’s take 729 as an example and follow the along with the steps:

Step 1: Recognize the prime number that can divide the given number.

In the case of 729, the prime factor is 3. So, we got our prime factor.

Step 2:Now, we need to keep dividing until the quotient becomes 1.

Number to be divided

Prime number divisor

Quotient

729

3

243

243

3

81

81

3

27

27

3

9

9

3

3

3

3

1

Step 3: Now, we are ready to figure out the square root. First, we count the number of times we divided the number with the divisor.

We have divided 729 by 3 for six times.

729 = 3 x 3 x 3 x 3 x 3 x 3

Step 4: Now, we consider the equal factors in a pair and multiply them. See how we are breaking 729 into tiny squares of its own.

729 = (3 x 3) x (3 x 3) x (3 x 3)

Note how each pair has two identical factors. That means we get a square number if we multiply those two factors. Note how we have already broken 729 into smaller square numbers.

Step 5: Square root the smaller square numbers and multiply. The product is the square root.

Therefore,

But what if the number is not a perfect square (numbers whose square roots are integers)? Let us take the example of 24

First, we prime factorize

Number to be divided

Prime factor divisor

Quotient

24

2

12

12

2

6

6

2

3

3

3

1

24 = 2 x 2 x 2 x 3

Next, we make our pairs

24 = (2 x 2) x (2 x 3)

It doesn’t look very similar, right? We have two unequal factors paired up. But the good news is that we can divide these numbers into their square roots.

24 = (2 x 2) x (√2 x √2) x (√3 x √3)

On dividing the right-hand side with the identical factor, we get:

2 x √2 x √3

Since numbers under roots can be multiplied, we can multiply √2 and √3. Multiplying two roots is the same as normal multiplication, but the two numbers that have been multiplied are under square roots

Prime factorization is a decent method to find square roots. However, when the numbers are large, it often becomes tiring.

This is where we use the long-division method. It is like a regular division, but there are a few differences in how we use the divisor. Let’s try to find the square root of 2025 using the long division method:

Step 1: Separate the number into two-digit parts

Here, we separate 2025 into 20 and 25.

Step 2: Subtract the closest square number lesser than the dividend, in our case, 16, from the first part, which is 20. Write down the number that has been squared, in our case, 4, on the left side and the top.

Step 3: Bring the second pair of digits down. Hence, we get 425 as the new dividend (number to be divided).

Step 4: This is the tricky part. First, multiply the quotient (here, 4) by 2. This is the first digit of our next divisor.

Note how two blanks have been left beside 8. To fill those blanks, we need to follow the next step.

Step 5: Next, we must fill the blanks with the same integers and multiply. The product is then subtracted from the dividend. We continue the process until the remainder hits 0 for a perfect square. The last digit of the new divisor is placed beside the quotient, here 4.. This number is the square root of 2025.

Hence, √2025 = 45

We can even find answers to numbers that aren’t perfect squares (square numbers whose roots aren’t whole numbers). Let’s take the example of 24:

Step 1: We need to subtract the closest square number, which has to be lower than 24, from the dividend 24 and put the roots of the square to the left and on top.

Step 2: There’s a remainder and no other pairs to work with. This is where we bring the decimal point. Adding a decimal point to the quotient allows us to add a pair of 0s beside the remainder. For every pair of 0’s added to the remainder, we put another pair beside the number we are square root. Let’s start by adding one pair and bringing it down.

Step 3: Then we multiply the value on the quotient by 2. We leave space for the second digit and follow along the steps.

Here, we can see that 8 is the last digit of the divisor 88. Hence, we put it beside the decimal in the quotient

Step 4: Now, it is time to add another pair of 0’s to the remainder and divide.

Step 5: In this step, we need to add the digit we placed in the blank with the divisor to the divisor. Here, we add 8 to 88. The sum gives us the first two digits of the next new divisor.

Step 6: Repeat steps 2 to 5 for each step; here, we have done till two places after the decimal.

You can go further and get a more accurate value. We have stopped at 4.89, which is reasonably accurate.

There’s an even faster way to find the square root of a number. Let’s see how that works.

Approximation method of finding square roots

The approximation method is suitable in cases where the difference from the actual value is negligible. Let’s take 24 as an example again.

The steps are as follows:

Find out the two perfect square numbers between which our number lies. Here, 24 lies between 16 (4^{2}) and 25 (5^{2}). Since 25 is greater than 24, the first digit of the square root of 24 is definitely 4.

To find the digits after the decimal, we need to use a formula:

Now, we have to put the numbers after the decimal. Here, we have already found that the root starts with 4. Therefore, the answer is 4.8888…, or 4.89 if rounded off.

Bonus: You won’t have to worry about squaring numbers that end in 5. A quick trick does all the work.

Take the example of 75.

We can multiply 75 by itself, but we can also use a quicker method to calculate the square numbers. First, put 25 in the place of the last two digits. The digits that come before are the product between the first digit of the original number and the digit adjacent to it.

For example: 75^{2 }= (7×8)25= 5625.

Mathematics is incomplete without squares and square roots. Many formulae in different subjects make use of square numbers. One such beautiful formula is the simple E=mc^{2}. It entered planetary sciences, statistics, trades and commerce, and even the music we enjoy. If you know how to find the square root of a number, you can speed up solving various equations.

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