Square Root of Decimals

A square root is simply a number that, when multiplied by itself, gives the original 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
Answer: 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 number, 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 number

• 2.50 has one digit 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.
  
  Practice Problem: Find the square root of 0.25
  Answer: 0.5
  Explanation:
  Here, the decimal is simple.
   

1. So, convert it into a fraction first.
    0.25 = 25 / 100
    
2. 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.
  
  Practice Problem: Calculate 0.5's square root.
  Answer: 0.707
  Explanation: 
   

1. Think about this: 
   √0.49 = 0.7 
   √0.64 = 0.8
    
2. 0.5 in this case falls between 0.49 and 0.64.
    
3. We are aware that 0.5's square root must be less than 0.8 but slightly greater than 0.7.
    
4. 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.
  
  Practice Problem: Find the square root of 0.2025
  Answer: 0.45​​​​​​​
  Explanation:
   

1. Put the numbers into the following pairs: (20) and (25)
    
2. 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.
    
3. Find the nearest square less than 20.
   4 × 4 = 16, which is less than twenty.
    
4. ​​​​​​​So, 20 − 16 = 4.
    
5. The following pair: 25
    
6. Now subtract 25 so that it is 425. 
    
7. Now double 4 = 8
    
8. Find a value of x such that
   (80 + x) x × ≤ 425
    
9. Let's try x = 5 
   (80 + 5) × 5 ≤ 425
    
10. 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:

Let's understand it with an example.

Example: Find the square root of 0.6
Answer: 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

1. Let us take 0.75
   Multiply by itself:
   0.75 × 0.75 = 0.5625
   So, it is less than 0.6
    
2. Now try 0.77
   Multiply by itself:
   0.77 × 0.77 = 0.5929
   Still, it is low.
    
3. 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
Answer: 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: 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. 
   

1. Let's use x:
   \((60+x)× x ≤ 396\)
    
2. Assume, x = 6
   (60 + 6) × 6 = 66 × 6 = 396
    
3. 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

Practice Problem: Find the square root of 7.21 by yourself.

Here are some tips and tricks to master square root of decimals:

1. To convert a number from decimal number system to fraction, multiply and divide the number with powers of 10. Here, the power will be equal to the number of digits after the decimal. 
    
2. Always simplify the fraction if needed.
    
3. Memorize the basic squares and square root of numbers.
    
4. You can use multiplication tables to find the product of numbers.
    
5. You can use a multiplication calculator to verify your answers.

Parent Tip: 

• Help your children mug up square and square roots of basic counting numbers.
• Encourage them to lean multiplications using multiplication tables and charts.
• Ask them to solve all given examples and practice problems by themselves.

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

1. 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.
   
    
2. Estimating travel distance in navigation
   The distance between two points with decimal coordinates is calculated by navigators.
   
   Example: The distance is \(\sqrt{(0.6^2 + 0.3^2)}=√0.45 ≅ 0.67082039325 \ km. \)
    
3. 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.
   
    
4. 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 \(\sqrt{(0.6^2+0.4^2)}=\sqrt{0.52}=0.7211m\)
   
    
5. 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, \(\sqrt{0.81}=0.9m\)