Understanding Square Root of a Decimal Number

Any number can be expressed as the product of the prime numbers. This method of representation of a number in terms of the product of prime numbers is termed as the prime factorization method. It is the easiest method known for the manual calculation of the square root of decimal numbers. But this method becomes tedious and tiresome when the amount involved is large. In order to beat this problem, we use the division method. 

Consider the following method for finding the square root of the decimal number. It is explained with the assistance of an example for a transparent understanding.

Note: The number of digits in a perfect square is very significant for calculating its square root of a decimal number by the long division method. 

What is Square Root of Decimal Number

The square root of decimals is calculated in the same way as the square root of whole numbers.

Inverse operations include taking the square root of a number and squaring a number. The square root of a number is the number that is multiplied by itself to give the original number, whereas the square of a number is the value of the number's power 2.

The value of a decimal number raised to the power 1/2 is called the square root of the decimal. The square root of 24.01, for example, is 4.9, as (4.9)2 = 24.01.

The estimation approach or the long division method can be used to calculate the square root of a decimal value.

The pairings of whole number parts and fractional parts are separated using bars in the long division method.

After that, long division is performed in the same manner as any other whole number.

Steps for Finding the Square Root of Decimal Number with Examples

Square Root: Estimation Method

Estimation and approximation are ways to make calculations easier and more realistic by making a good guess of the real value. 

This method can also help you figure out and approximate the square root of a number you're given. 

We only need to find the perfect square numbers that are closest to the given decimal number to figure out its approximate square root value.

Let's find the square root of 31.36. Below are the steps:

1. Find the perfect square numbers that are closest to 31.36.

2. The perfect square numbers closest to 31.36 are 25 and 36.

3. √25 = 5 and √36 = 6. This means that √31.36 is somewhere between 5 and 6.

4. Now, we must determine if √31.36 is closer to 5 or 6. 

5. Let’s consider 5.5 and 6.

6. 5.52 = 30.25 and 62= 36. As a result, √31.36 is near to 5.5 and lies between 5.5 and 6.

As a result, the square root of 31.36 is around 5.5.

Square Root: Long Division Method

When we need to divide big numbers into steps or parts, we utilize the long division approach, which breaks the division problem down into a series of simpler steps. 

Using this strategy, we may get the precise square root of any number.

Let's find the square root of 2.56. Below are the steps:

1. Place a bar over each pair of digits starting with the unit. We will have two pairs, i.e. 2 and 56.

2. Then divide it by the biggest number whose square is less than or equal to it.

3. Here, the whole number part is 2 and we have 1 x 1 = 1. So, the quotient is 1.

4. Reducing the number, that is, the pair of fractions under the remainder bar (that is 1).

5. Add the quotient's last digit to the divisor, which is 1 + 1 = 2. Find a suitable number to the right of the obtained sum (that is 2) that, when combined with the sum's result, provides a new divisor for the new dividend (that is, 156) that is brought down. As we get closer to the fractional part, add a decimal after 1 in the quotient.

6. The new quotient number will be the same as the divisor number, therefore, the divisor will now be 26 and the quotient will be 1.6 because 26 x 6 = 156. (The criterion is the same — the dividend must be less than or equal to it.)

7. Using a decimal point, continue the process by adding zeros in pairs to the remainder.

8. The resultant quotient is the square root of the number. As a result, 2.56 has a square root of 1.6.

Examples to be Solved 

Question 1: Find the square root of decimal number 29.16.

Solution: The following steps will explain how to find the square root of decimal number 29.16 by using the long division method: 

Step 1. Write down the decimal number and then make pairs of the integer and fractional parts separately. Then, the pair of the integers of a decimal number is created from right to left and so, the pair of the fractional part is made right from the start of the decimal point.

Example: Within the decimal number 29.16, 29 is one pair and then 16 is another pair.

Step 2. Find the amount whose pair is a smaller amount than or adequate to the primary pair. In the number 29.16, 5’s square is adequate to 25. Hence, we'll write 5 within the divisor and 5 within the quotient.

Step 3. Now, we will subtract 25 from 29. The answer is 4. We will bring down the opposite pair which is 16 and put the percentage point within the quotient.

Step 4. Now, we'll multiply the divisor by 2. Since 5 into 2 is adequate to 10, so we'll write 10 below the divisor. We need to seek out the third digit of the amount in order that it's completely divisible by the amount 416. We already have two digits 10. The 3rd digit should be 4 because 104 . 4 = 116.

 Step 5. Write 4 in the quotient's place. Hence, the answer is 5.4.

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Question 2: Find the square root of decimal number 84.64 by using a long division method. 

Solution: Follow these steps to seek out the root of this decimal number 84.64.

Step 1. Write down the decimal number and then make pairs of the integer and fractional parts separately. The pair of the integers of a decimal number is made from right to left and so the pair of the fractional part is made right from the start of the decimal point.

So, in the decimal number 84.64, 84 is one pair and then 64 is another one.

Step 2. Find the amount whose pair is a smaller amount than or adequate to the primary pair. In the number 84.64, 9’s square is equal to 81. Hence, we'll write 9 within the divisor and 9 within the quotient.

Step 3. Now, we'll subtract 81 from 84. The answer is 3. We will bring down the opposite pair which is 64 and put the percentage point within the quotient after 9.

Step 4. Now, we'll multiply the divisor by 2. Since 9 into 2 is adequate to 18, so we'll write 18  below the divisor. We have to find out the third digit for the number so that it is totally divisible by the number 364. We already have two digits 18. The 3rd digit should be 2 because 182 . 2 = 364.

Step 5. Write 2 within the quotient's place after the percentage point. Hence, the answer is 9.2.