How to Find the 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:
Find the perfect square numbers that are closest to 31.36.
The perfect square numbers closest to 31.36 are 25 and 36.
√25 = 5 and √36 = 6. This means that √31.36 is somewhere between 5 and 6.
Now, we must determine if √31.36 is closer to 5 or 6.
Let’s consider 5.5 and 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:
Place a bar over each pair of digits starting with the unit. We will have two pairs, i.e. 2 and 56.
Then divide it by the biggest number whose square is less than or equal to it.
Here, the whole number part is 2 and we have 1 x 1 = 1. So, the quotient is 1.
Reducing the number, that is, the pair of fractions under the remainder bar (that is 1).
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.
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.)
Using a decimal point, continue the process by adding zeros in pairs to the remainder.
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.
FAQs on Square Root of Decimal Numbers: Complete Guide
1. What is a decimal number, and how does the concept of its square root differ from that of a whole number?
A decimal number consists of a whole number part and a fractional part, separated by a decimal point. While the fundamental concept of a square root (a value that, when multiplied by itself, gives the original number) remains the same for both whole numbers and decimals, the primary difference lies in the calculation method. Finding the square root of a decimal requires careful handling of the decimal point, for which the long division method is specifically adapted.
2. What is the most reliable method for finding the square root of any decimal number?
The long division method is the most reliable and universally applicable technique for finding the square root of a decimal number. Unlike methods such as prime factorization, which are impractical for non-integer values, the long division method provides a step-by-step algorithm that works for both perfect squares (like 6.25) and non-perfect squares (like 5.3), allowing you to calculate the root to any desired number of decimal places.
3. Can you explain the steps to find the square root of a decimal using the long division method?
Certainly. To find the square root of a decimal number using the long division method, follow these steps as per the CBSE Class 8 syllabus:
Step 1: Pair the Digits: Starting from the decimal point, pair the digits of the whole number part from right to left. Then, pair the digits of the decimal part from left to right. If there is an odd number of decimal digits, add a zero at the end to make a pair.
Step 2: Perform Division: Find the largest number whose square is less than or equal to the first pair. This number is the first digit of your quotient and the divisor.
Step 3: Place the Decimal Point: Once you have finished with the whole number part and need to bring down the first pair after the decimal point, place a decimal point in the quotient.
Step 4: Continue Division: Bring down the next pair of digits and continue the long division process until you achieve the desired precision.
4. How do you find the square root of a number like 2.56 as an example?
To find the square root of 2.56, we use the long division method. First, pair the digits: the whole number part is '2', and the decimal part is '56'. Find a number whose square is ≤ 2, which is 1. Subtract 1 from 2, leaving 1. Bring down the pair '56' to get 156. Place the decimal point in the quotient. Double the quotient (1), which gives 2. Now, find a digit 'x' such that 2x multiplied by x is 156. We find that 26 × 6 = 156. Therefore, the square root of 2.56 is 1.6.
5. Why is the long division method generally preferred over prime factorization for decimal numbers?
The long division method is preferred because it is a direct and universal algorithm. Prime factorization requires converting the decimal into a fraction (e.g., 2.56 = 256/100) and then finding the square root of the numerator and denominator separately. This is only feasible if both are perfect squares. The long division method, however, works systematically for any decimal number, perfect square or not, without needing to convert it to a fraction, making it far more practical and efficient.
6. What is the most important rule to remember about placing the decimal point in the answer?
The most critical rule is that the decimal point in the quotient (the answer) must be placed directly above the decimal point in the number you are finding the square root of. This is done precisely at the step when you have dealt with the entire whole number part and are about to bring down the first pair of digits from the decimal part. Incorrect placement is a very common error.
7. Where are square roots of decimal numbers used in real-world applications?
The square root of decimal numbers has many practical applications where precision is key. For example:
Construction and Architecture: To calculate the side length of a square plot of land or room when its area is given in decimal units, like 42.25 square metres.
Physics: In formulas from kinematics or mechanics, such as calculating the radius of a circle from its area (A = πr²) or finding the time period of a pendulum, where measurements are rarely whole numbers.
Statistics: To calculate the standard deviation, a measure of data spread, for datasets involving decimal values.
8. If a perfect square number has '2n' decimal places, how many decimal places will its square root have?
If a perfect square number has '2n' decimal places, its square root will have exactly 'n' decimal places. The number of decimal places is halved when you take the square root. For instance, the square root of a number with 4 decimal places (like 0.0625) will have 2 decimal places (0.25). This is a direct consequence of the pairing of digits in the long division method, where each pair of decimal digits yields one decimal digit in the root.
