Square Root of Decimals

A square root is simply the number that was multiplied by itself to get another number. It is the inverse of squaring. While squaring multiplies a number by itself, taking the square root works in reverse and finds the original value. We show square roots using the √ symbol. The result you get after solving is called the root.

Example:
Find the Square Root of 64
Here, we are asked to find:
√64
The Answer is: 8

Explanation:
We need to find which number, when multiplied by itself, gives 64.
8 × 8 = 64
So, 8 is the square root of 64.
 

Decimals are numbers that, when separated by a decimal point, represent both whole numbers and portions of whole numbers. Values smaller than one, such as tenths, hundredths, and thousandths, are represented by the digits to the right of the decimal point, while the whole number is indicated by the digits to the left. 
Example:
The whole number,

• 2.50 has two digits before the decimal point.
• The fractional number is 50 after the decimal point.
• In 2.50, the decimal point separates the two whole numbers from the fractional parts.
   

A decimal's square root is the value that yields the original decimal number when multiplied by itself. It can be located using techniques like estimation, long division, or the recognition of well-known square root patterns.

Example:
Find the square root of 0.36
Solution:
Here,
We require two numbers that add up to 0.36 when multiplied by one another.
Let’s try with 0.6:
0.6 × 0.6 = 0.36
So, the final Answer: 
√0.36 = 0.6
 

Finding the square root of a decimal follows the same ideas as with whole numbers.
Here are some simple ways to do it:

Convert the Decimal into a Fraction

When the decimal is simple, convert it into a fraction. Then find the square root of the numerator and the denominator separately.
Example:
Find the square root of 0.25
The Answer is: 0.5
Explanation:
Here, the decimal is simple.
So, convert it into a fraction first.
 0.25 = 25 / 100
Now take the square root of both the top and the bottom:
√0.25 = √(25 / 100) = √25 / √100 = 5 / 10 = 0.5

Estimate and Adjust

We cannot readily obtain an exact square root when the decimal is not a perfect square. Therefore, we estimate it by comparing it to square roots.
Example:
Calculate 0.5's square root.
The Answer is 0.707
Explanation:
Think about this: 
√0.49 = 0.7 
√0.64 = 0.8
0.5 in this case falls between 0.49 and 0.64.
We are aware that 0.5's square root must be less than 0.8 but slightly greater than 0.7.
Let's estimate: 0.71 × 0.71 = 0.5041 (quite near).
√0.5 ≅ 0.707.

Long Division Method

The Long division method helps you find the square root of a decimal, one digit at a time.
Example:
Find the square root of 0.2025
The Answer is 0.45
Explanation:
Put the numbers into the following pairs: (20) and (25)
Since the number is less than 1, the square root will also be less than 1. Therefore, the answer needs to include the decimal point.

Find the nearest square less than 20.
4 × 4 = 16, which is less than twenty.
So, 20 − 16 = 4.

The following pair: 25
Now subtract 25 so that it is 425. 
Now double 4 = 8
Find a value of x such that
(80 + x) x × ≤ 425
Let's try x = 5 
(80 + 5) × 5 ≤ 425
so, √0.2025 = 0.45.

The estimation method helps you find the square root of a decimal by using nearby perfect squares. Here are some simple steps for it:
Example:
Find the square root of 0.6
The Answer is 0.775
Explanation:

Step 1:
Take the nearby perfect squares
Let us consider,
√0.49 = 0.7
√0.64 = 0.8
So, √0.6 must be between 0.7 and 0.8

Step 2:
Assume a number between 0.7 and 0.8
Let us take 0.75
           Multiply by itself:
           0.75 × 0.75 = 0.5625
So, it is less than 0.6
Now try 0.77
Multiply by itself:
0.77 × 0.77 = 0.5929
Still, it is low.
Try with 0.775
Multiply by itself:
0.775 × 0.775 = 0.600625
So, it is close to 0.6.
The Estimated square root,
          √0.6 ≅ 0.774596669
 

When a decimal number is not a perfect square, the long division method is a conventional and precise method to determine its square root. Here are a few easy steps to follow:
Example:
Find the square root of 12.96
The Answer is 3.6
Explanation:
Step 1:
Group digits into pairs:
(12) and (96)
Step 2:
Find the number that should be less than or equal to 12, when multiplied by itself.
3 × 3 = 9 (less than 12)
So, here we get 3 as the first digit answer.
Step 3:
Now,
Subtract 9 from 12.
12 − 9 = 3
Bring down the 96 so we get 396.
Step 4:
Double the number we’ve found so far:
3 × 2 = 6
Now use this 6 as the start of the next divisor.
Step 5:
Find the following number now. 
Let's use x:
(60+x)× x ≤ 396
Assume, x = 6
(60 + 6) × 6 = 66 × 6 = 396
Now, 
Enter 6 as the following number in the response.
The square root is 3.6.
Step 6:
Subtract 396 − 396 = 0
√12.96 = 3.6

You might think the square root of a decimal only lives in math books, but they are also used in everyday life.

Determining lens focal length in photography  
The length of a focal lens combination is calculated using the square root of the combined optical power.
Example:
To measure the length of the lens, 0.64 diopters. √0.64 = 0.8 diopters help adjust focus for clear images.

Estimating travel distance in navigation
The distance between two points with decimal coordinates is calculated by navigators.
Example:
The distance is √(0.62 + 0.32)=√0.45 ≅ 0.67082039325 km.

Calculating the side length of a square plot
The side length of a square plot is measured.
Example:
The area of a square plot (0.36 m2) requires the square root.√0.36=0.6m , which helps people plan accordingly.

Optimizing material cuts in carpentry
It calculates the diagonal of a rectangular wooden panel.
Example:
To find the Diagonal
0.4m by 0.6 and the diagonal is √(0.62+0.42)=√0.52=0.7211m

Assessing structural load distribution
In engineering, the square root of the decimal area determines the side length of a square foundation.
Example:
Area is 0.81m2 so, √0.81=0.9m